Theorem dfcomember 39783

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 39777	. 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
2		df-coels 39528	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
3	2	erALTVeq1i 39781	. 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴)
4	1, 3	bitr4i 281	1 ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   E cep 5549  ^◡ccnv 5649   ↾ cres 5652   ≀ ccoss 39209   ∼ ccoels 39210   ErALTV werALTV 39235   CoMembEr wcomember 39239

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8723  df-qs 8727  df-coels 39528  df-refrel 39618  df-symrel 39650  df-trrel 39684  df-eqvrel 39695  df-dmqs 39749  df-erALTV 39775  df-comember 39777

This theorem is used by:  dfcomember2  39784