Theorem dferALTV2 37533

Description: Equivalence relation with natural domain predicate, see the comment of df-ers 37528. (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 37529	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 37504	. . 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 274	1 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541  dom cdm 5676   / cqs 8701   EqvRel weqvrel 37055   DomainQs wdmqs 37062   ErALTV werALTV 37064

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 397  df-dmqs 37504  df-erALTV 37529

This theorem is referenced by:  erALTVeq1  37534  dfcomember2  37538  erimeq  37544  partim  37673  pet0  37680  petid  37682  petidres  37684  petinidres  37686  petxrnidres  37688  mainer  37699  petincnvepres  37714  pet  37716