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Theorem dfcomember 38674
Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)
Assertion
Ref Expression
dfcomember ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Proof of Theorem dfcomember
StepHypRef Expression
1 df-comember 38668 . 2 ( CoMembEr 𝐴 ↔ ≀ ( E ↾ 𝐴) ErALTV 𝐴)
2 df-coels 38414 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32erALTVeq1i 38672 . 2 ( ∼ 𝐴 ErALTV 𝐴 ↔ ≀ ( E ↾ 𝐴) ErALTV 𝐴)
41, 3bitr4i 278 1 ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206   E cep 5582  ccnv 5683  cres 5686  ccoss 38183  ccoels 38184   ErALTV werALTV 38209   CoMembEr wcomember 38211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748  df-qs 8752  df-coels 38414  df-refrel 38514  df-symrel 38546  df-trrel 38576  df-eqvrel 38587  df-dmqs 38641  df-erALTV 38666  df-comember 38668
This theorem is referenced by:  dfcomember2  38675
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