Theorem dfcomember 38649

Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.)

Assertion

Ref	Expression
dfcomember	⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Proof of Theorem dfcomember

Step	Hyp	Ref	Expression
1		df-comember 38643	. 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
2		df-coels 38388	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	erALTVeq1i 38647	. 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
4	1, 3	bitr4i 278	1 ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

Syntax hints:   ↔ wb 206   E cep 5522  ^◡ccnv 5622   ↾ cres 5625   ≀ ccoss 38154   ∼ ccoels 38155   ErALTV werALTV 38180   CoMembEr wcomember 38182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8634  df-qs 8638  df-coels 38388  df-refrel 38488  df-symrel 38520  df-trrel 38550  df-eqvrel 38561  df-dmqs 38615  df-erALTV 38641  df-comember 38643

This theorem is referenced by:  dfcomember2  38650