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Theorem dfcoeleqvrel 37113
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 37110, eqvrelcoss3 37109 and eqvrelcoss4 37111 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 37078 . 2 ( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
2 df-coels 36903 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32eqvreleqi 37094 . 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 5541  ccnv 5637  cres 5640  ccoss 36663  ccoels 36664   EqvRel weqvrel 36680   CoElEqvRel wcoeleqvrel 36682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-coels 36903  df-refrel 37003  df-symrel 37035  df-trrel 37065  df-eqvrel 37076  df-coeleqvrel 37078
This theorem is referenced by:  dfcomember3  37165
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