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Theorem dmqs1cosscnvepreseq 38644
Description: Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.)
Assertion
Ref Expression
dmqs1cosscnvepreseq ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)

Proof of Theorem dmqs1cosscnvepreseq
StepHypRef Expression
1 df-coels 38394 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
21dmqseqeq1i 38626 . 2 ((dom ∼ 𝐴 /𝐴) = 𝐴 ↔ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴)
3 dmqscoelseq 38643 . 2 ((dom ∼ 𝐴 /𝐴) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
42, 3bitr3i 277 1 ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537   cuni 4912   E cep 5588  ccnv 5688  dom cdm 5689  cres 5691   / cqs 8743  ccoss 38162  ccoels 38163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750  df-coss 38393  df-coels 38394
This theorem is referenced by:  eqvreldmqs  38657  eqvreldmqs2  38658  eldisjn0el  38788
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