Theorem dfcoeleqvrel 38146

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 38143, eqvrelcoss3 38142 and eqvrelcoss4 38144 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 38111	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
2		df-coels 37936	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	eqvreleqi 38127	. 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 3	bitr4i 277	1 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

Syntax hints:   ↔ wb 205   E cep 5576  ^◡ccnv 5672   ↾ cres 5675   ≀ ccoss 37701   ∼ ccoels 37702   EqvRel weqvrel 37718   CoElEqvRel wcoeleqvrel 37720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-coels 37936  df-refrel 38036  df-symrel 38068  df-trrel 38098  df-eqvrel 38109  df-coeleqvrel 38111

This theorem is referenced by:  dfcomember3  38198