Theorem dferALTV2 38646

Description: Equivalence relation with natural domain predicate, see the comment of df-ers 38641. (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 38642	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 38616	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 623	. 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 1540  dom cdm 5619   / cqs 8624   EqvRel weqvrel 38172   DomainQs wdmqs 38179   ErALTV werALTV 38181

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 38616  df-erALTV 38642

This theorem is referenced by:  erALTVeq1  38647  dfcomember2  38651  erimeq  38657  partim  38786  pet0  38793  petid  38795  petidres  38797  petinidres  38799  petxrnidres  38801  mainer  38812  petincnvepres  38827  pet  38829