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Theorem dfcomember 38055
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 38049 . 2 ( CoMembEr 𝐴 ↔ ≀ ( E ↾ 𝐴) ErALTV 𝐴)
2 df-coels 37795 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32erALTVeq1i 38053 . 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 205   E cep 5572  ccnv 5668  cres 5671  ccoss 37556  ccoels 37557   ErALTV werALTV 37582   CoMembEr wcomember 37584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-qs 8711  df-coels 37795  df-refrel 37895  df-symrel 37927  df-trrel 37957  df-eqvrel 37968  df-dmqs 38022  df-erALTV 38047  df-comember 38049
This theorem is referenced by:  dfcomember2  38056
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