Theorems · Theorem · logic and foundations

FirstOrder.Language.Substructure.mem_closed_iff

∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] (s : Set M),
  s ∈ (FirstOrder.Language.Substructure.closure L).closed ↔
    ∀ {n : ℕ} (f : L.Functions n), FirstOrder.Language.ClosedUnder f s

Defined in: Mathlib.ModelTheory.Substructures
Cited by: 1 results in Mathlib
Foundations: Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: FirstOrder.Language.Structure

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Cites14

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 and proof · cited by 8,199
FirstOrder.Languagestatement and proof · cited by 1,084
FirstOrder.Language.Structurestatement and proof · cited by 775
FirstOrder.Language.Substructurestatement · cited by 242
FirstOrder.Language.Functionsstatement and proof · cited by 153
LowerAdjoint.toFunproof · cited by 105
reflproof · cited by 80
FirstOrder.Language.Substructure.closurestatement and proof · cited by 70
FirstOrder.Language.ClosedUnderstatement and proof · cited by 13
FirstOrder.Language.Substructure.subset_closureproof · cited by 12
LowerAdjoint.closedstatement and proof · cited by 9

Cited by1

Results whose statement or proof uses this declaration.

FirstOrder.Language.Substructure.mem_closed_of_isRelationalproof · cited by 2