Theorem dfcoeleqvrel 39038

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 39035, eqvrelcoss3 39034 and eqvrelcoss4 39036 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 39003	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
2		df-coels 38834	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	eqvreleqi 39019	. 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 3	bitr4i 278	1 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

Syntax hints:   ↔ wb 206   E cep 5521  ^◡ccnv 5621   ↾ cres 5624   ≀ ccoss 38515   ∼ ccoels 38516   EqvRel weqvrel 38532   CoElEqvRel wcoeleqvrel 38534

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-coels 38834  df-refrel 38924  df-symrel 38956  df-trrel 38990  df-eqvrel 39001  df-coeleqvrel 39003

This theorem is referenced by:  dfcomember3  39091