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Theorem dmqscoelseq 36074
 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
dmqscoelseq ((dom ∼ 𝐴 /𝐴) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)

Proof of Theorem dmqscoelseq
StepHypRef Expression
1 dmcoels 35876 . . 3 dom ∼ 𝐴 = 𝐴
21qseq1i 35725 . 2 (dom ∼ 𝐴 /𝐴) = ( 𝐴 /𝐴)
32eqeq1i 2803 1 ((dom ∼ 𝐴 /𝐴) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  ∪ cuni 4801  dom cdm 5520   / cqs 8274   ∼ ccoels 35633 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-eprel 5431  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-qs 8281  df-coss 35838  df-coels 35839 This theorem is referenced by:  dmqs1cosscnvepreseq  36075  dfmember3  36087
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