Theorem dfcoeleqvrel 39778

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 39775, eqvrelcoss3 39774 and eqvrelcoss4 39776 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 39743	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
2		df-coels 39574	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	eqvreleqi 39759	. 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 3	bitr4i 281	1 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   E cep 5553  ^◡ccnv 5653   ↾ cres 5656   ≀ ccoss 39255   ∼ ccoels 39256   EqvRel weqvrel 39272   CoElEqvRel wcoeleqvrel 39274

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-coels 39574  df-refrel 39664  df-symrel 39696  df-trrel 39730  df-eqvrel 39741  df-coeleqvrel 39743

This theorem is used by:  dfcomember3  39831