Theorem dferALTV2 39120

Description: Equivalence relation with natural domain predicate, see the comment of df-ers 39115. (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 39116	. 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴))
2		df-dmqs 39090	. . 3 ⊢ (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
3	2	anbi2i 629	. 2 ⊢ (( EqvRel 𝑅 ∧ 𝑅 DomainQs 𝐴) ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
4	1, 3	bitri 276	1 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547  dom cdm 5618   / cqs 8632   EqvRel weqvrel 38567   DomainQs wdmqs 38574   ErALTV werALTV 38576

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 208  df-an 397  df-dmqs 39090  df-erALTV 39116

This theorem is referenced by:  erALTVeq1  39121  dfcomember2  39125  erimeq  39131  partim  39278  pet0  39285  petid  39287  petidres  39289  petinidres  39291  petxrnidres  39293  mainer  39315  petincnvepres  39330  pet  39332