Theorem dferALTV2 39004

Description: Equivalence relation with natural domain predicate, see the comment of df-ers 38999. (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 39000	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 38974	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 624	. 2 ⊢ (( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4	1, 3	bitri 275	1 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  dom cdm 5632   / cqs 8644   EqvRel weqvrel 38451   DomainQs wdmqs 38458   ErALTV werALTV 38460

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 207  df-an 396  df-dmqs 38974  df-erALTV 39000

This theorem is referenced by:  erALTVeq1  39005  dfcomember2  39009  erimeq  39015  partim  39162  pet0  39169  petid  39171  petidres  39173  petinidres  39175  petxrnidres  39177  mainer  39199  petincnvepres  39214  pet  39216