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Theorem dferALTV2 38049
Description: Equivalence relation with natural domain predicate, see the comment of df-ers 38044. (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 38045 . 2 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅𝑅 DomainQs 𝐴))
2 df-dmqs 38020 . . 3 (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
32anbi2i 622 . 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 395   = wceq 1533  dom cdm 5669   / cqs 8701   EqvRel weqvrel 37571   DomainQs wdmqs 37578   ErALTV werALTV 37580
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 396  df-dmqs 38020  df-erALTV 38045
This theorem is referenced by:  erALTVeq1  38050  dfcomember2  38054  erimeq  38060  partim  38189  pet0  38196  petid  38198  petidres  38200  petinidres  38202  petxrnidres  38204  mainer  38215  petincnvepres  38230  pet  38232
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