Theorem dfcoeleqvrel 38728

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 38725, eqvrelcoss3 38724 and eqvrelcoss4 38726 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 38693	. 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
2		df-coels 38524	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	eqvreleqi 38709	. 2 ⊢ ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
4	1, 3	bitr4i 278	1 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)

Syntax hints:   ↔ wb 206   E cep 5513  ^◡ccnv 5613   ↾ cres 5616   ≀ ccoss 38232   ∼ ccoels 38233   EqvRel weqvrel 38249   CoElEqvRel wcoeleqvrel 38251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-coels 38524  df-refrel 38614  df-symrel 38646  df-trrel 38680  df-eqvrel 38691  df-coeleqvrel 38693

This theorem is referenced by:  dfcomember3  38782