Theorems · Definition · logic and foundations
FirstOrder.Language.Equiv.symm
{L : FirstOrder.Language} →
{M : Type w} → {N : Type w'} → [inst : L.Structure M] → [inst_1 : L.Structure N] → L.Equiv M N → L.Equiv N MThe inverse of a first-order equivalence is a first-order equivalence.
- Defined in
- Mathlib.ModelTheory.Basic
- Cited by
- 32 results in Mathlib
- Foundations
- Depth 18 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.
- Equivproof · cited by 8,337
- Equiv.symmproof · cited by 3,681
- FirstOrder.Languagestatement and proof · cited by 1,084
- FirstOrder.Language.Structurestatement and proof · cited by 775
- FirstOrder.Language.Functionsproof · cited by 153
- FirstOrder.Language.Relationsproof · cited by 147
- FirstOrder.Language.Equivstatement and proof · cited by 83
- FirstOrder.Language.Equiv.toEquivproof · cited by 12
Cited by36
Results whose statement or proof uses this declaration.
- FirstOrder.Language.PartialEquiv.symmproof · cited by 9
- FirstOrder.Language.Equiv.apply_symm_applystatement · cited by 6
- FirstOrder.Language.DirectLimit.partialEquivLimitproof · cited by 6
- FirstOrder.Language.PartialEquiv.cod_le_codproof · cited by 6
- FirstOrder.Language.PartialEquiv.toEmbeddingOfEqTopproof · cited by 5
- FirstOrder.Language.Equiv.fg_iffproof · cited by 3
- FirstOrder.Language.Equiv.self_comp_symmstatement and proof · cited by 3
- FirstOrder.Language.Equiv.symm_apply_applystatement · cited by 3
- FirstOrder.Language.Hereditary.is_equiv_invariant_of_fgproof · cited by 2
- FirstOrder.Language.PartialEquiv.toEquivOfEqTopproof · cited by 2
- FirstOrder.Language.age.countable_quotientproof · cited by 2
- FirstOrder.Language.Equiv.symm_comp_selfstatement and proof · cited by 2