Theorem dfcomember 38628

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 38622	. 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
2		df-coels 38368	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	erALTVeq1i 38626	. 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 5598  ^◡ccnv 5699   ↾ cres 5702   ≀ ccoss 38135   ∼ ccoels 38136   ErALTV werALTV 38161   CoMembEr wcomember 38163

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ec 8765  df-qs 8769  df-coels 38368  df-refrel 38468  df-symrel 38500  df-trrel 38530  df-eqvrel 38541  df-dmqs 38595  df-erALTV 38620  df-comember 38622

This theorem is referenced by:  dfcomember2  38629