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Theorem dfcomember2 38654
Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)
Assertion
Ref Expression
dfcomember2 ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))

Proof of Theorem dfcomember2
StepHypRef Expression
1 dfcomember 38653 . 2 ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
2 dferALTV2 38649 . 2 ( ∼ 𝐴 ErALTV 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))
31, 2bitri 275 1 ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  dom cdm 5688   / cqs 8742  ccoels 38162   EqvRel weqvrel 38178   ErALTV werALTV 38187   CoMembEr wcomember 38189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ec 8745  df-qs 8749  df-coels 38393  df-refrel 38493  df-symrel 38525  df-trrel 38555  df-eqvrel 38566  df-dmqs 38620  df-erALTV 38645  df-comember 38647
This theorem is referenced by:  dfcomember3  38655
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