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Theorem eqvreldmqs 38654
Description: Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)
Assertion
Ref Expression
eqvreldmqs (( EqvRel ≀ ( E ↾ 𝐴) ∧ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))

Proof of Theorem eqvreldmqs
StepHypRef Expression
1 df-coeleqvrel 38566 . . 3 ( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
21bicomi 224 . 2 ( EqvRel ≀ ( E ↾ 𝐴) ↔ CoElEqvRel 𝐴)
3 dmqs1cosscnvepreseq 38641 . 2 ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
42, 3anbi12i 628 1 (( EqvRel ≀ ( E ↾ 𝐴) ∧ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540   cuni 4905   E cep 5581  ccnv 5682  dom cdm 5683  cres 5685   / cqs 8740  ccoss 38160  ccoels 38161   EqvRel weqvrel 38177   CoElEqvRel wcoeleqvrel 38179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  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 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-eprel 5582  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-ec 8743  df-qs 8747  df-coss 38390  df-coels 38391  df-coeleqvrel 38566
This theorem is referenced by:  mpet3  38815
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