Theorem dferALTV2 38140

Description: Equivalence relation with natural domain predicate, see the comment of df-ers 38135. (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 38136	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 38111	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 622	. 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 205   ∧ wa 395   = wceq 1534  dom cdm 5678   / cqs 8724   EqvRel weqvrel 37665   DomainQs wdmqs 37672   ErALTV werALTV 37674

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 206  df-an 396  df-dmqs 38111  df-erALTV 38136

This theorem is referenced by:  erALTVeq1  38141  dfcomember2  38145  erimeq  38151  partim  38280  pet0  38287  petid  38289  petidres  38291  petinidres  38293  petxrnidres  38295  mainer  38306  petincnvepres  38321  pet  38323