Theorems · Theorem · logic and foundations
FirstOrder.Language.Substructure.closure_le
∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {S : L.Substructure M} {s : Set M},
(FirstOrder.Language.Substructure.closure L).toFun s ≤ S ↔ s ⊆ ↑SA substructure S includes closure L s if and only if it includes s.
- Defined in
- Mathlib.ModelTheory.Substructures
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- FirstOrder.Languagestatement and proof · cited by 1,084
- FirstOrder.Language.Structurestatement and proof · cited by 775
- FirstOrder.Language.Substructurestatement and proof · cited by 242
- LowerAdjoint.toFunstatement · cited by 105
- FirstOrder.Language.Substructure.closurestatement and proof · cited by 70
- FirstOrder.Language.Substructure.closedproof · cited by 1
- LowerAdjoint.closure_le_closed_iff_leproof · cited by 1
Cited by7
Results whose statement or proof uses this declaration.
- FirstOrder.Language.Substructure.map_closureproof · cited by 4
- FirstOrder.Language.Substructure.closure_inductionproof · cited by 3
- FirstOrder.Language.equiv_between_cgproof · cited by 2
- FirstOrder.Language.age_directLimitproof · cited by 1
- FirstOrder.Language.Hom.eqOn_closureproof · cited by 1
- FirstOrder.Language.embedding_from_cgproof · cited by 0