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Theorem dferALTV2 38669
Description: Equivalence relation with natural domain predicate, see the comment of df-ers 38664. (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 38665 . 2 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅𝑅 DomainQs 𝐴))
2 df-dmqs 38640 . . 3 (𝑅 DomainQs 𝐴 ↔ (dom 𝑅 / 𝑅) = 𝐴)
32anbi2i 623 . 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 206  wa 395   = wceq 1540  dom cdm 5685   / cqs 8744   EqvRel weqvrel 38199   DomainQs wdmqs 38206   ErALTV werALTV 38208
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 38640  df-erALTV 38665
This theorem is referenced by:  erALTVeq1  38670  dfcomember2  38674  erimeq  38680  partim  38809  pet0  38816  petid  38818  petidres  38820  petinidres  38822  petxrnidres  38824  mainer  38835  petincnvepres  38850  pet  38852
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