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Theorem dfcomember 37537
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 37531 . 2 ( CoMembEr 𝐴 ↔ ≀ ( E ↾ 𝐴) ErALTV 𝐴)
2 df-coels 37277 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32erALTVeq1i 37535 . 2 ( ∼ 𝐴 ErALTV 𝐴 ↔ ≀ ( E ↾ 𝐴) ErALTV 𝐴)
41, 3bitr4i 277 1 ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   E cep 5579  ccnv 5675  cres 5678  ccoss 37038  ccoels 37039   ErALTV werALTV 37064   CoMembEr wcomember 37066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8704  df-qs 8708  df-coels 37277  df-refrel 37377  df-symrel 37409  df-trrel 37439  df-eqvrel 37450  df-dmqs 37504  df-erALTV 37529  df-comember 37531
This theorem is referenced by:  dfcomember2  37538
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