Mathlib Map

Theorems · Theorem · logic and foundations

FirstOrder.Language.MeetsDefinable.closure_eq_self

∀ {L : FirstOrder.Language} {M : Type u_1} [inst : L.Structure M] {A : Set M},
  FirstOrder.Language.MeetsDefinable A → ↑((FirstOrder.Language.Substructure.closure L).toFun A) = A

The closure of a set meeting definable sets is equal to itself.

Defined in
Mathlib.ModelTheory.ElementarySubstructures
Cited by
1 results in Mathlib
Foundations
Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
FirstOrder.Language.Structure
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

FirstOrder.Language.MeetsDefinable.isElementary_closure · cited by 0MeetsDefinable.isElementa…Set · cited by 53352SetSetLike.coe · cited by 8199SetLike.coeSet.ofPred · cited by 6101Set.ofPredSet.range · cited by 4705Set.rangeFirstOrder.Language · cited by 1084FirstOrder.LanguageFirstOrder.Language.Structure · cited by 775Language.StructureFirstOrder.Language.Substructure · cited by 242Language.SubstructureSet.Subset.antisymm · cited by 213Subset.antisymmFirstOrder.Language.Term · cited by 166Language.TermLowerAdjoint.toFun · cited by 105LowerAdjoint.toFunFirstOrder.Language.Term.realize · cited by 81Term.realizeFirstOrder.Language.Formula.Realize · cited by 81Formula.RealizeFirstOrder.Language.Substructure.closure · cited by 70Substructure.closureSet.singleton_nonempty · cited by 32Set.singleton_nonemptyFirstOrder.Language.Term.relabel · cited by 27Term.relabelMeetsDefinable.closure_eq_selfCITED BYCITES

Cites24

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

Cited by1

Results whose statement or proof uses this declaration.