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Theorem disjdmqs 37669
Description: If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 37672 and petlem 37677 via disjdmqseq 37670. (Contributed by Peter Mazsa, 16-Sep-2021.)
Assertion
Ref Expression
disjdmqs ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 /𝑅))

Proof of Theorem disjdmqs
StepHypRef Expression
1 disjdmqsss 37667 . 2 ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 /𝑅))
2 disjdmqscossss 37668 . 2 ( Disj 𝑅 → (dom ≀ 𝑅 /𝑅) ⊆ (dom 𝑅 / 𝑅))
31, 2eqssd 3999 1 ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 /𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  dom cdm 5676   / cqs 8701  ccoss 37038   Disj wdisjALTV 37072
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8704  df-qs 8708  df-coss 37276  df-cnvrefrel 37392  df-disjALTV 37570
This theorem is referenced by:  disjdmqseq  37670
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