Theorem dferALTV2 39292

Description: Equivalence relation with natural domain predicate, see the comment of df-ers 39287. (Contributed by Peter Mazsa, 26-Jun-2021.) (Revised by Peter Mazsa, 30-Aug-2021.)

Assertion

Ref	Expression
dferALTV2	⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Proof of Theorem dferALTV2

Step	Hyp	Ref	Expression
1		df-erALTV 39288	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 39262	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 634	. 2 ⊢ (( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4	1, 3	bitri 278	1 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  dom cdm 5662   / cqs 8693   EqvRel weqvrel 38739   DomainQs wdmqs 38746   ErALTV werALTV 38748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-dmqs 39262  df-erALTV 39288

This theorem is referenced by:  erALTVeq1  39293  dfcomember2  39297  erimeq  39303  partim  39450  pet0  39457  petid  39459  petidres  39461  petinidres  39463  petxrnidres  39465  mainer  39487  petincnvepres  39502  pet  39504