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Theorem dfcoeleqvrel 36358
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 36355, eqvrelcoss3 36354 and eqvrelcoss4 36356 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 36323 . 2 ( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
2 df-coels 36161 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32eqvreleqi 36339 . 2 ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
41, 3bitr4i 281 1 ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209   E cep 5433  ccnv 5524  cres 5527  ccoss 35956  ccoels 35957   EqvRel weqvrel 35973   CoElEqvRel wcoeleqvrel 35975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-coels 36161  df-refrel 36253  df-symrel 36281  df-trrel 36311  df-eqvrel 36321  df-coeleqvrel 36323
This theorem is referenced by:  dfmember3  36409
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