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Theorem dfcomember3 38676
Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
Assertion
Ref Expression
dfcomember3 ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))

Proof of Theorem dfcomember3
StepHypRef Expression
1 dfcomember2 38675 . 2 ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))
2 dfcoeleqvrel 38624 . . . 4 ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
32bicomi 224 . . 3 ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴)
4 dmqscoelseq 38663 . . 3 ((dom ∼ 𝐴 /𝐴) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
53, 4anbi12i 628 . 2 (( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
61, 5bitri 275 1 ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539   cuni 4906  dom cdm 5684   / cqs 8745  ccoels 38184   EqvRel weqvrel 38200   CoElEqvRel wcoeleqvrel 38202   CoMembEr wcomember 38211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-eprel 5583  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748  df-qs 8752  df-coss 38413  df-coels 38414  df-refrel 38514  df-symrel 38546  df-trrel 38576  df-eqvrel 38587  df-coeleqvrel 38589  df-dmqs 38641  df-erALTV 38666  df-comember 38668
This theorem is referenced by:  mainer  38836  mpet  38841
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