Theorem dfcoeleqvrel 37487

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 37484, eqvrelcoss3 37483 and eqvrelcoss4 37485 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)

Assertion

Ref	Expression
dfcoeleqvrel	⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

Proof of Theorem dfcoeleqvrel

Step	Hyp	Ref	Expression
1		df-coeleqvrel 37452	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
2		df-coels 37277	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	eqvreleqi 37468	. 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 3	bitr4i 277	1 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

Syntax hints:   ↔ wb 205   E cep 5579  ^◡ccnv 5675   ↾ cres 5678   ≀ ccoss 37038   ∼ ccoels 37039   EqvRel weqvrel 37055   CoElEqvRel wcoeleqvrel 37057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-coels 37277  df-refrel 37377  df-symrel 37409  df-trrel 37439  df-eqvrel 37450  df-coeleqvrel 37452

This theorem is referenced by:  dfcomember3  37539