Theorems · Theorem · logic and foundations
FirstOrder.Language.Substructure.subset_closure
∀ {L : FirstOrder.Language} {M : Type w} [inst : L.Structure M] {s : Set M},
s ⊆ ↑((FirstOrder.Language.Substructure.closure L).toFun s)The substructure generated by a set includes the set.
- Defined in
- Mathlib.ModelTheory.Substructures
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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 · cited by 242
- LowerAdjoint.toFunstatement · cited by 105
- FirstOrder.Language.Substructure.closurestatement and proof · cited by 70
- LowerAdjoint.le_closureproof · cited by 2
Cited by12
Results whose statement or proof uses this declaration.
- FirstOrder.Language.Substructure.map_closureproof · cited by 4
- FirstOrder.Language.exists_elementarySubstructure_card_eqproof · cited by 1
- FirstOrder.Language.MeetsDefinable.closure_eq_selfproof · cited by 1
- FirstOrder.Language.Substructure.CG.of_map_embeddingproof · cited by 1
- FirstOrder.Language.Substructure.mem_closed_iffproof · cited by 1
- FirstOrder.Language.Substructure.FG.of_map_embeddingproof · cited by 1
- FirstOrder.Language.isUltrahomogeneous_iff_IsExtensionPairproof · cited by 1
- FirstOrder.Language.IsFraisseLimit.isExtensionPairproof · cited by 1
- FirstOrder.Language.Substructure.notMem_of_notMem_closureproof · cited by 0
- FirstOrder.Language.Substructure.closure_induction'statement and proof · cited by 0
- FirstOrder.Language.Substructure.closure_withConstants_eqstatement and proof · cited by 0