Theorem dfcomember2 39882

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 39881	. 2 ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
2		dferALTV2 39877	. 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 5655   / cqs 8744   ∼ ccoels 39308   EqvRel weqvrel 39324   ErALTV werALTV 39333   CoMembEr wcomember 39337

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8747  df-qs 8751  df-coels 39626  df-refrel 39716  df-symrel 39748  df-trrel 39782  df-eqvrel 39793  df-dmqs 39847  df-erALTV 39873  df-comember 39875

This theorem is used by:  dfcomember3  39883