Metamath Proof Explorer

Theorem dferALTV2

Description: Equivalence relation with natural domain predicate, see the comment of df-ers . (Contributed by Peter Mazsa, 26-Jun-2021) (Revised by Peter Mazsa, 30-Aug-2021)

Ref Expression
Assertion dferALTV2 ⊢ R ErALTV A ↔ EqvRel R ∧ dom ⁡ R / R = A

Proof

Step Hyp Ref Expression
1 df-erALTV ⊢ R ErALTV A ↔ EqvRel R ∧ R DomainQs A
2 df-dmqs ⊢ R DomainQs A ↔ dom ⁡ R / R = A
3 2 anbi2i ⊢ EqvRel R ∧ R DomainQs A ↔ EqvRel R ∧ dom ⁡ R / R = A
4 1 3 bitri ⊢ R ErALTV A ↔ EqvRel R ∧ dom ⁡ R / R = A