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Theorem dfcoeleqvrel 36735
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 36732, eqvrelcoss3 36731 and eqvrelcoss4 36733 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 36700 . 2 ( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
2 df-coels 36538 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
32eqvreleqi 36716 . 2 ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
41, 3bitr4i 277 1 ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   E cep 5494  ccnv 5588  cres 5591  ccoss 36333  ccoels 36334   EqvRel weqvrel 36350   CoElEqvRel wcoeleqvrel 36352
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-coels 36538  df-refrel 36630  df-symrel 36658  df-trrel 36688  df-eqvrel 36698  df-coeleqvrel 36700
This theorem is referenced by:  dfmember3  36786
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