Theorem dferALTV2 39252

Description: Equivalence relation with natural domain predicate, see the comment of df-ers 39247. (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 39248	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 39222	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 632	. 2 ⊢ (( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4	1, 3	bitri 277	1 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1560  dom cdm 5647   / cqs 8677   EqvRel weqvrel 38699   DomainQs wdmqs 38706   ErALTV werALTV 38708

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 209  df-an 400  df-dmqs 39222  df-erALTV 39248

This theorem is referenced by:  erALTVeq1  39253  dfcomember2  39257  erimeq  39263  partim  39410  pet0  39417  petid  39419  petidres  39421  petinidres  39423  petxrnidres  39425  mainer  39447  petincnvepres  39462  pet  39464