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Theorem disjdmqs 37543
Description: If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 37546 and petlem 37551 via disjdmqseq 37544. (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 37541 . 2 ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 /𝑅))
2 disjdmqscossss 37542 . 2 ( Disj 𝑅 → (dom ≀ 𝑅 /𝑅) ⊆ (dom 𝑅 / 𝑅))
31, 2eqssd 3996 1 ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 /𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  dom cdm 5670   / cqs 8687  ccoss 36912   Disj wdisjALTV 36946
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 5293  ax-nul 5300  ax-pr 5421
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 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-ec 8690  df-qs 8694  df-coss 37150  df-cnvrefrel 37266  df-disjALTV 37444
This theorem is referenced by:  disjdmqseq  37544
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