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 ) )