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Theorem dfcoeleqvrel 38018
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 38015, eqvrelcoss3 38014 and eqvrelcoss4 38016 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 37983 . 2 ( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
2 df-coels 37808 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32eqvreleqi 37999 . 2 ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
41, 3bitr4i 278 1 ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   E cep 5575  ccnv 5671  cres 5674  ccoss 37570  ccoels 37571   EqvRel weqvrel 37587   CoElEqvRel wcoeleqvrel 37589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-coels 37808  df-refrel 37908  df-symrel 37940  df-trrel 37970  df-eqvrel 37981  df-coeleqvrel 37983
This theorem is referenced by:  dfcomember3  38070
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