Theorem dfcomember2 38056

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 38055	. 2 ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
2		dferALTV2 38051	. 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
3	1, 2	bitri 275	1 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533  dom cdm 5669   / cqs 8704   ∼ ccoels 37557   EqvRel weqvrel 37573   ErALTV werALTV 37582   CoMembEr wcomember 37584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-qs 8711  df-coels 37795  df-refrel 37895  df-symrel 37927  df-trrel 37957  df-eqvrel 37968  df-dmqs 38022  df-erALTV 38047  df-comember 38049

This theorem is referenced by:  dfcomember3  38057