Theorem dferALTV2 39422

Description: Equivalence relation with natural domain predicate, see the comment of df-ers 39417. (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 39418	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 39392	. . 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 1570  dom cdm 5661   / cqs 8689   EqvRel weqvrel 38869   DomainQs wdmqs 38876   ErALTV werALTV 38878

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 39392  df-erALTV 39418

This theorem is referenced by:  erALTVeq1  39423  dfcomember2  39427  erimeq  39433  partim  39580  pet0  39587  petid  39589  petidres  39591  petinidres  39593  petxrnidres  39595  mainer  39617  petincnvepres  39632  pet  39634