diff --git a/tla-pp/src/main/scala/at/forsyte/apalache/tla/pp/SetMembershipSimplifier.scala b/tla-pp/src/main/scala/at/forsyte/apalache/tla/pp/SetMembershipSimplifier.scala
index e0faf5917c..771b6a0570 100644
--- a/tla-pp/src/main/scala/at/forsyte/apalache/tla/pp/SetMembershipSimplifier.scala
+++ b/tla-pp/src/main/scala/at/forsyte/apalache/tla/pp/SetMembershipSimplifier.scala
@@ -7,9 +7,18 @@ import at.forsyte.apalache.tla.lir.transformations.{LanguageWatchdog, Transforma
 import at.forsyte.apalache.tla.lir.values._
 
 /**
- * A simplifier that rewrites vacuously true membership tests based on type information.
+ * A simplifier that rewrites expressions commonly found in `TypeOK`. Assumes expressions to be well-typed.
  *
- * For example, `x \in BOOLEAN` is rewritten to `TRUE` if x is typed BoolT1.
+ * After Apalache's type-checking, we can rewrite some expressions to simpler forms. For example, the (after
+ * type-checking) vacuously true `x \in BOOLEAN` is rewritten to `TRUE` (as `x` must be a `BoolT1`).
+ *
+ * We currently perform the following simplifications (for type-defining sets TDS, see [[isTypeDefining]]):
+ *   - `n \in Nat` ~> `x >= 0`
+ *   - `b \in BOOLEAN`, `i \in Int`, `r \in Real` ~> `TRUE`
+ *   - `seq \in Seq(TDS)` ~> `TRUE`
+ *   - `set \in SUBSET TDS` ~> `TRUE`
+ *   - `fun \in [TDS1 -> TDS2]` ~> `TRUE`
+ *   - `fun \in [Dom -> TDS]` ~> `DOMAIN fun = Dom`
  *
  * @author
  *   Thomas Pani
@@ -27,46 +36,57 @@ class SetMembershipSimplifier(tracker: TransformationTracker) extends AbstractTr
   }
 
   /**
-   * Returns the type of a TLA+ predefined set, if rewriting set membership to TRUE is applicable. In particular, it is
-   * *not* applicable to `Nat`, since `i \in Nat` does not hold for all `IntT1`-typed `i`.
+   * Returns true iff the passed TLA+ expression is a type-defining set. Type-defining sets contain all of the values of
+   * their respective element type.
+   *
+   * The type-defining sets are inductively defined as
+   *   - the predefined sets BOOLEAN, Int, Real, STRING,
+   *   - sets of sequences over type-defining sets, e.g., Seq(BOOLEAN), Seq(Int), Seq(Seq(Int)), Seq(SUBSET Int), ...
+   *   - power sets of type-defining sets, e.g., SUBSET BOOLEAN, SUBSET Int, SUBSET Seq(Int), ...
+   *   - sets of functions over type-defining sets, e.g., [Int -> BOOLEAN], ...
+   *
+   * In particular, `Nat` is not type-defining, nor are sequence sets / power sets thereof, since `i \in Nat` does not
+   * hold for all `IntT1`-typed `i`.
    */
-  private def typeOfSupportedPredefSet: PartialFunction[TlaPredefSet, TlaType1] = {
-    case TlaBoolSet => BoolT1()
-    case TlaIntSet  => IntT1()
-    case TlaRealSet => RealT1()
-    case TlaStrSet  => StrT1()
-    // intentionally omits TlaNatSet, see above.
-  }
+  private def isTypeDefining: Function[TlaEx, Boolean] = {
+    // base case: BOOLEAN, Int, Real, STRING
+    case ValEx(TlaBoolSet) | ValEx(TlaIntSet) | ValEx(TlaRealSet) | ValEx(TlaStrSet) => true
 
-  /**
-   * Checks if this transformation is applicable (see [[typeOfSupportedPredefSet]]) to a TLA+ predefined set `ps` of
-   * primitives, and if the types of `name` and `ps` match.
-   */
-  private def isApplicable(name: TlaEx, ps: TlaPredefSet): Boolean =
-    typeOfSupportedPredefSet.isDefinedAt(ps) && name.typeTag == Typed(typeOfSupportedPredefSet(ps))
+    // inductive cases:
+    // 1. Seq(s) for a type-defining set `s`
+    case OperEx(TlaSetOper.seqSet, set) => isTypeDefining(set)
+    // 2. SUBSET s for a type-defining set `s`
+    case OperEx(TlaSetOper.powerset, set) => isTypeDefining(set)
+    // 3. [s1 -> s2] for type-defining sets `s1` and `s2
+    case OperEx(TlaSetOper.funSet, set1, set2) => isTypeDefining(set1) && isTypeDefining(set2)
 
-  /**
-   * Checks if this transformation is applicable (see [[typeOfSupportedPredefSet]]) to a TLA+ predefined set of
-   * sequences (`Seq(_)`) `ps`, and if the types of `name` and `ps` match.
-   */
-  private def isApplicableSeq(name: TlaEx, ps: TlaPredefSet): Boolean =
-    typeOfSupportedPredefSet.isDefinedAt(ps) && name.typeTag == Typed(SeqT1(typeOfSupportedPredefSet(ps)))
+    // otherwise
+    case _ => false
+  }
 
   /**
-   * Rewrites vacuously true membership tests based on type information, and rewrites `i \in Nat` to `i \ge 0`.
+   * Simplifies expressions commonly found in `TypeOK`, assuming they are well-typed.
    *
-   * For example, `x \in BOOLEAN` is rewritten to `TRUE` if `x` is typed `BoolT1`.
+   * @see
+   *   [[SetMembershipSimplifier]] for a full list of supported rewritings.
    */
   private def transformMembership: PartialFunction[TlaEx, TlaEx] = {
-    // n \in Nat  ->  x >= 0
-    case OperEx(TlaSetOper.in, name, ValEx(TlaNatSet)) if name.typeTag == Typed(IntT1()) =>
-      OperEx(TlaArithOper.ge, name, ValEx(TlaInt(0))(intTag))(boolTag)
-    // b \in BOOLEAN, i \in Int, r \in Real  ->  TRUE
-    case OperEx(TlaSetOper.in, name, ValEx(ps: TlaPredefSet)) if isApplicable(name, ps) => trueVal
-    // seq \in Seq(_)  ->  TRUE
-    case OperEx(TlaSetOper.in, name, OperEx(TlaSetOper.seqSet, ValEx(ps: TlaPredefSet))) if isApplicableSeq(name, ps) =>
-      trueVal
-    // return `ex` unchanged
+    case ex @ OperEx(TlaSetOper.in, name, set) =>
+      set match {
+        // x \in TDS  ~>  TRUE
+        case set if isTypeDefining(set) => trueVal
+
+        // n \in Nat  ~>  x >= 0
+        case ValEx(TlaNatSet) => OperEx(TlaArithOper.ge, name, ValEx(TlaInt(0))(intTag))(boolTag)
+
+        // fun \in [Dom -> TDS]  ~>  DOMAIN fun = Dom    (fun \in [TDS1 -> TDS2] is handled above)
+        case OperEx(TlaSetOper.funSet, domain, set2) if isTypeDefining(set2) =>
+          OperEx(TlaOper.eq, OperEx(TlaFunOper.domain, name)(domain.typeTag), domain)(boolTag)
+
+        // otherwise, return `ex` unchanged
+        case _ => ex
+      }
+    // return non-set membership expressions unchanged
     case ex => ex
   }
 }
diff --git a/tla-pp/src/test/scala/at/forsyte/apalache/tla/pp/TestSetMembershipSimplifier.scala b/tla-pp/src/test/scala/at/forsyte/apalache/tla/pp/TestSetMembershipSimplifier.scala
index 2e1e3e730e..299836f6e3 100644
--- a/tla-pp/src/test/scala/at/forsyte/apalache/tla/pp/TestSetMembershipSimplifier.scala
+++ b/tla-pp/src/test/scala/at/forsyte/apalache/tla/pp/TestSetMembershipSimplifier.scala
@@ -28,6 +28,7 @@ class TestSetMembershipSimplifier
   private val boolName = tla.name("b").as(BoolT1())
   private val strName = tla.name("s").as(StrT1())
   private val intName = tla.name("i").as(IntT1())
+  private val funName = tla.name("fun").as(FunT1(IntT1(), BoolT1()))
 
   private val boolSet = tla.booleanSet().as(SetT1(BoolT1()))
   private val strSet = tla.stringSet().as(SetT1(StrT1()))
@@ -46,6 +47,19 @@ class TestSetMembershipSimplifier
   private val intSeqSet = tla.seqSet(tla.intSet()).as(SetT1(SeqT1(IntT1())))
   private val natSeqSet = tla.seqSet(tla.natSet()).as(SetT1(SeqT1(IntT1())))
 
+  private val boolSetVal = tla.enumSet(boolVal, boolName).as(SetT1(BoolT1()))
+  private val strSetVal = tla.enumSet(strVal, strName).as(SetT1(StrT1()))
+  private val intSetVal = tla.enumSet(intVal, intName).as(SetT1(IntT1()))
+
+  private val boolSetName = tla.name("boolSet").as(SetT1(BoolT1()))
+  private val strSetName = tla.name("strSet").as(SetT1(StrT1()))
+  private val intSetName = tla.name("intSet").as(SetT1(IntT1()))
+
+  private val boolPowerset = tla.seqSet(tla.booleanSet()).as(SetT1(SeqT1(BoolT1())))
+  private val strPowerset = tla.seqSet(tla.stringSet()).as(SetT1(SeqT1(StrT1())))
+  private val intPowerset = tla.seqSet(tla.intSet()).as(SetT1(SeqT1(IntT1())))
+  private val natPowerset = tla.seqSet(tla.natSet()).as(SetT1(SeqT1(IntT1())))
+
   val expressions = List(
       (boolName, boolVal, boolSet),
       (strName, strVal, strSet),
@@ -53,6 +67,9 @@ class TestSetMembershipSimplifier
       (boolSeqName, boolSeqVal, boolSeqSet),
       (strSeqName, strSeqVal, strSeqSet),
       (intSeqName, intSeqVal, intSeqSet),
+      (boolSetName, boolSetVal, boolPowerset),
+      (strSetName, strSetVal, strPowerset),
+      (intSetName, intSetVal, intPowerset),
   )
 
   override def beforeEach(): Unit = {
@@ -60,40 +77,90 @@ class TestSetMembershipSimplifier
   }
 
   test("simplifies appropriately-typed set membership") {
+    // i \in Nat  ~>  i >= 0
+    val intNameInNat = tla.in(intName, tla.natSet()).as(BoolT1())
+    val intValInNat = tla.in(intVal, tla.natSet()).as(BoolT1())
+    simplifier(intNameInNat) shouldBe tla.ge(intName, tla.int(0)).as(BoolT1())
+    simplifier(intValInNat) shouldBe tla.ge(intVal, tla.int(0)).as(BoolT1())
+
+    /* *** tests for all supported types of applicable sets *** */
+
     expressions.foreach { case (name, value, set) =>
+      // name \in ApplicableSet  ~>  TRUE
+      // e.g., b \in BOOLEAN, i \in Int, ...  ~>  TRUE
       val inputName = tla.in(name, set).as(BoolT1())
       simplifier(inputName) shouldBe tlaTrue
 
+      // literal \in ApplicableSet  ~>  TRUE
+      // e.g., TRUE \in BOOLEAN, 42 \in Int, ...  ~>  TRUE
       val inputValue = tla.in(value, set).as(BoolT1())
       simplifier(inputValue) shouldBe tlaTrue
 
+      /* *** nested cases *** */
+
+      // name \in ApplicableSet /\ _  ~>  TRUE
+      // e.g., i \in Int /\ _, ...  ~>  TRUE
       val nestedInputName = tla.and(tla.in(name, set).as(BoolT1()), tlaTrue).as(BoolT1())
       simplifier(nestedInputName) shouldBe tla.and(tlaTrue, tlaTrue).as(BoolT1())
 
+      // literal \in ApplicableSet /\ _  ~>  TRUE
+      // e.g., 42 \in Int /\ _, ...  ~>  TRUE
       val nestedInputValue = tla.and(tla.in(name, set).as(BoolT1()), tlaTrue).as(BoolT1())
       simplifier(nestedInputValue) shouldBe tla.and(tlaTrue, tlaTrue).as(BoolT1())
 
-      val intNameInNat = tla.in(intName, tla.natSet()).as(BoolT1())
-      val intValInNat = tla.in(intVal, tla.natSet()).as(BoolT1())
-      simplifier(intNameInNat) shouldBe tla.ge(intName, tla.int(0)).as(BoolT1())
-      simplifier(intValInNat) shouldBe tla.ge(intVal, tla.int(0)).as(BoolT1())
-    }
-  }
-
-  test("returns inappropriately-typed set membership unchanged") {
-    expressions.foreach { case (name, value, _) =>
-      expressions.filter { case (name2, _, _) => name != name2 }.foreach { case (_, _, otherSet) =>
-        val inputName = tla.in(name, otherSet).as(BoolT1())
-        simplifier(inputName) shouldBe inputName
-
-        val inputValue = tla.in(value, otherSet).as(BoolT1())
-        simplifier(inputValue) shouldBe inputValue
+      // fun \in [ApplicableSet1 -> ApplicableSet2], ...  ~>  TRUE
+      expressions.foreach { case (name2, _, set2) =>
+        val funSetType = SetT1(FunT1(name.typeTag.asTlaType1(), name2.typeTag.asTlaType1()))
+        val funInFunSet = tla.in(funName, tla.funSet(set, set2).as(funSetType)).as(BoolT1())
+        simplifier(funInFunSet) shouldBe tlaTrue
       }
     }
 
+    /* *** tests of particular expressions *** */
+
+    // <<{{1}}>> \in Seq(SUBSET Int)  ~>  TRUE
+    val setOfSetOfInt = SetT1(SetT1(IntT1()))
+    val seqOfSetOfSetOfInt = SeqT1(setOfSetOfInt)
+    val nestedSeqSubsetVal =
+      tla.tuple(tla.enumSet(intSetVal).as(setOfSetOfInt)).as(SeqT1(setOfSetOfInt)).as(seqOfSetOfSetOfInt)
+    val nestedSeqSubsetTest =
+      tla.in(nestedSeqSubsetVal, tla.seqSet(tla.powSet(intSet).as(setOfSetOfInt)).as(seqOfSetOfSetOfInt)).as(BoolT1())
+    simplifier(nestedSeqSubsetTest) shouldBe tlaTrue
+
+    // {<<1>>} \in SUBSET (Seq(Int))  ~>  TRUE
+    val setOfSeqOfInt = SetT1(SeqT1(IntT1()))
+    val nestedSubsetSeqVal = tla.enumSet(tla.tuple(intVal).as(SeqT1(IntT1()))).as(setOfSeqOfInt)
+    val nestedSubsetSeqTest = tla.in(nestedSubsetSeqVal, tla.powSet(intSeqSet).as(setOfSeqOfInt)).as(BoolT1())
+    simplifier(nestedSubsetSeqTest) shouldBe tlaTrue
+
+    // fun \in [Seq(SUBSET Int) -> SUBSET Seq(BOOLEAN)], ...  ~>  TRUE
+    val intPowersetSeqType = SeqT1(SetT1(IntT1()))
+    val boolSeqPowersetType = SetT1(SeqT1(BoolT1()))
+    val nestedFunSetType = SetT1(FunT1(intPowersetSeqType, boolSeqPowersetType))
+    val nestedInput = tla
+      .in(funName,
+          tla
+            .funSet(tla.seqSet(intSeqSet).as(intPowersetSeqType), tla.powSet(boolSeqSet).as(boolSeqPowersetType))
+            .as(nestedFunSetType))
+      .as(BoolT1())
+    simplifier(nestedInput) shouldBe tlaTrue
+
+    // fun \in [RM -> PredefSet], ...  ~>  DOMAIN fun = RM
+    val domain = tla.name("RM").as(SetT1(IntT1()))
+    val funSetType = SetT1(FunT1(BoolT1(), IntT1()))
+    val funConstToBoolean = tla.in(funName, tla.funSet(domain, boolSet).as(funSetType)).as(BoolT1())
+    simplifier(funConstToBoolean) shouldBe tla.eql(tla.dom(funName).as(SetT1(IntT1())), domain).as(BoolT1())
+  }
+
+  test("returns myInt \\in Nat unchanged") {
     val intSeqNameInSeqNat = tla.in(intSeqName, natSeqSet).as(BoolT1())
     val intSeqValInSeqNat = tla.in(intSeqVal, natSeqSet).as(BoolT1())
     simplifier(intSeqNameInSeqNat) shouldBe intSeqNameInSeqNat
     simplifier(intSeqValInSeqNat) shouldBe intSeqValInSeqNat
+
+    val intSetNameInNatPowerset = tla.in(intSetName, natPowerset).as(BoolT1())
+    val intSetValInNatPowerset = tla.in(intSetVal, natPowerset).as(BoolT1())
+    simplifier(intSetNameInNatPowerset) shouldBe intSetNameInNatPowerset
+    simplifier(intSetValInNatPowerset) shouldBe intSetValInNatPowerset
   }
 }