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Theorem disjdmqs 38308
Description: If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 38311 and petlem 38316 via disjdmqseq 38309. (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 38306 . 2 ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 /𝑅))
2 disjdmqscossss 38307 . 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 1533  dom cdm 5682   / cqs 8730  ccoss 37681   Disj wdisjALTV 37715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rmo 3374  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-qs 8737  df-coss 37915  df-cnvrefrel 38031  df-disjALTV 38209
This theorem is referenced by:  disjdmqseq  38309
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