Loogle!

Result

Found 10 declarations mentioning ZFSet.iUnion.

• ZFSet.iUnion 📋 Mathlib.SetTheory.ZFC.Basic
  {α : Type u_1} [Small.{u, u_1} α] (f : α → ZFSet.{u}) : ZFSet.{u}
• ZFSet.subset_iUnion 📋 Mathlib.SetTheory.ZFC.Basic
  {α : Type u_1} [Small.{u, u_1} α] (f : α → ZFSet.{u}) (i : α) : f i ⊆ ZFSet.iUnion fun i => f i
• ZFSet.coe_iUnion 📋 Mathlib.SetTheory.ZFC.Basic
  {α : Type u_1} [Small.{u, u_1} α] (f : α → ZFSet.{u}) : ↑(ZFSet.iUnion fun i => f i) = ⋃ i, ↑(f i)
• ZFSet.mem_iUnion 📋 Mathlib.SetTheory.ZFC.Basic
  {α : Type u_1} [Small.{u, u_1} α] {f : α → ZFSet.{u}} {x : ZFSet.{u}} : (x ∈ ZFSet.iUnion fun i => f i) ↔ ∃ i, x ∈ f i
• ZFSet.toSet_iUnion 📋 Mathlib.SetTheory.ZFC.Basic
  {α : Type u_1} [Small.{u, u_1} α] (f : α → ZFSet.{u}) : ↑(ZFSet.iUnion fun i => f i) = ⋃ i, ↑(f i)
• ZFSet.lift_card_iUnion_le_sum_card 📋 Mathlib.SetTheory.ZFC.Cardinal
  {α : Type u} [Small.{v, u} α] {f : α → ZFSet.{v}} : Cardinal.lift.{u, v} (ZFSet.iUnion fun i => f i).card ≤ Cardinal.sum fun i => (f i).card
• ZFSet.iSup_card_le_card_iUnion 📋 Mathlib.SetTheory.ZFC.Cardinal
  {α : Type u} [Small.{v, u} α] {f : α → ZFSet.{v}} : ⨆ i, (f i).card ≤ (ZFSet.iUnion fun i => f i).card
• ZFSet.rank_iUnion 📋 Mathlib.SetTheory.ZFC.Rank
  {α : Type u_1} [Small.{u, u_1} α] (f : α → ZFSet.{u}) : (ZFSet.iUnion fun i => f i).rank = ⨆ i, (f i).rank
• ZFSet.IsTransitive.iUnion 📋 Mathlib.SetTheory.ZFC.Ordinal
  {α : Type u_1} [Small.{u, u_1} α] {f : α → ZFSet.{u}} (hf : ∀ (i : α), (f i).IsTransitive) : (ZFSet.iUnion fun i => f i).IsTransitive
• ZFSet.vonNeumann_of_isSuccPrelimit 📋 Mathlib.SetTheory.ZFC.VonNeumann
  {o : Ordinal.{u}} (h : Order.IsSuccPrelimit o) : ZFSet.vonNeumann o = ZFSet.iUnion fun a => ZFSet.vonNeumann ↑a

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 401c76f serving mathlib revision a3d2529