Theorem dfcomember2 39138

Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)

Assertion

Ref	Expression
dfcomember2	⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))

Proof of Theorem dfcomember2

Step	Hyp	Ref	Expression
1		dfcomember 39137	. 2 ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
2		dferALTV2 39133	. 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
3	1, 2	bitri 277	1 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:   ↔ wb 208   ∧ wa 397   = wceq 1548  dom cdm 5620   / cqs 8636   ∼ ccoels 38564   EqvRel weqvrel 38580   ErALTV werALTV 38589   CoMembEr wcomember 38593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ec 8639  df-qs 8643  df-coels 38882  df-refrel 38972  df-symrel 39004  df-trrel 39038  df-eqvrel 39049  df-dmqs 39103  df-erALTV 39129  df-comember 39131

This theorem is referenced by:  dfcomember3  39139