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Theorem dferALTV2 37180
Description: Equivalence relation with natural domain predicate, see the comment of df-ers 37175. (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
StepHypRef Expression
1 df-erALTV 37176 . 2 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅𝑅 DomainQs 𝐴))
2 df-dmqs 37151 . . 3 (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
32anbi2i 624 . 2 (( EqvRel 𝑅𝑅 DomainQs 𝐴) ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
41, 3bitri 275 1 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  dom cdm 5637   / cqs 8653   EqvRel weqvrel 36701   DomainQs wdmqs 36708   ErALTV werALTV 36710
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 398  df-dmqs 37151  df-erALTV 37176
This theorem is referenced by:  erALTVeq1  37181  dfcomember2  37185  erimeq  37191  partim  37320  pet0  37327  petid  37329  petidres  37331  petinidres  37333  petxrnidres  37335  mainer  37346  petincnvepres  37361  pet  37363
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