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Theorem dfcomember 38200
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 38194 . 2 ( CoMembEr 𝐴 ↔ ≀ ( E ↾ 𝐴) ErALTV 𝐴)
2 df-coels 37940 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32erALTVeq1i 38198 . 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 5575  ccnv 5671  cres 5674  ccoss 37705  ccoels 37706   ErALTV werALTV 37731   CoMembEr wcomember 37733
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 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8725  df-qs 8729  df-coels 37940  df-refrel 38040  df-symrel 38072  df-trrel 38102  df-eqvrel 38113  df-dmqs 38167  df-erALTV 38192  df-comember 38194
This theorem is referenced by:  dfcomember2  38201
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