Theorems · Theorem · logic and foundations

FirstOrder.Language.Substructure.closure_iUnion

∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {ι : Sort u_3} (s : ι → Set M),
  (FirstOrder.Language.Substructure.closure L).toFun (⋃ i, s i) =
    ⨆ i, (FirstOrder.Language.Substructure.closure L).toFun (s i)

Defined in: Mathlib.ModelTheory.Substructures
Cited by: 3 results in Mathlib
Foundations: Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: FirstOrder.Language.Structure

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Cites12

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement and proof · cited by 53,352
SetLike.coestatement · cited by 8,199
Set.iUnionstatement · cited by 2,483
iSupstatement · cited by 2,415
FirstOrder.Languagestatement and proof · cited by 1,084
FirstOrder.Language.Structurestatement and proof · cited by 775
FirstOrder.Language.Substructurestatement · cited by 242
GaloisInsertion.gcproof · cited by 137
LowerAdjoint.toFunstatement · cited by 105
GaloisConnection.l_iSupproof · cited by 78
FirstOrder.Language.Substructure.closurestatement · cited by 70
FirstOrder.Language.Substructure.giproof · cited by 4

Cited by3

Results whose statement or proof uses this declaration.

FirstOrder.Language.Substructure.mem_iSup_of_directedproof · cited by 1
FirstOrder.Language.DirectLimit.cgproof · cited by 1
FirstOrder.Language.Substructure.iSup_eq_closureproof · cited by 0