Theorem dfcomember2 39610

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 39609	. 2 ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
2		dferALTV2 39605	. 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
3	1, 2	bitri 278	1 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  dom cdm 5647   / cqs 8694   ∼ ccoels 39036   EqvRel weqvrel 39052   ErALTV werALTV 39061   CoMembEr wcomember 39065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  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 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ec 8697  df-qs 8701  df-coels 39354  df-refrel 39444  df-symrel 39476  df-trrel 39510  df-eqvrel 39521  df-dmqs 39575  df-erALTV 39601  df-comember 39603

This theorem is used by:  dfcomember3  39611