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Theorem eqvreldmqs 38618
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 38530 . . 3 ( CoElEqvRel 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
21bicomi 224 . 2 ( EqvRel ≀ ( E ↾ 𝐴) ↔ CoElEqvRel 𝐴)
3 dmqs1cosscnvepreseq 38605 . 2 ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
42, 3anbi12i 627 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 1535   cuni 4914   E cep 5581  ccnv 5682  dom cdm 5683  cres 5685   / cqs 8737  ccoss 38122  ccoels 38123   EqvRel weqvrel 38139   CoElEqvRel wcoeleqvrel 38141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  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 8740  df-qs 8744  df-coss 38354  df-coels 38355  df-coeleqvrel 38530
This theorem is referenced by:  mpet3  38779
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