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Theorem dfcoeleqvrel 38604
Description: Alternate definition of the coelement equivalence relation predicate: a coelement equivalence relation is an equivalence relation on coelements. Other alternate definitions should be based on eqvrelcoss2 38601, eqvrelcoss3 38600 and eqvrelcoss4 38602 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
Assertion
Ref Expression
dfcoeleqvrel ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

Proof of Theorem dfcoeleqvrel
StepHypRef Expression
1 df-coeleqvrel 38569 . 2 ( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
2 df-coels 38394 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32eqvreleqi 38585 . 2 ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
41, 3bitr4i 278 1 ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206   E cep 5588  ccnv 5688  cres 5691  ccoss 38162  ccoels 38163   EqvRel weqvrel 38179   CoElEqvRel wcoeleqvrel 38181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-coels 38394  df-refrel 38494  df-symrel 38526  df-trrel 38556  df-eqvrel 38567  df-coeleqvrel 38569
This theorem is referenced by:  dfcomember3  38656
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