Theorem disjdmqs 38187

Description: If a relation is disjoint, its domain quotient is equal to the domain quotient of the cosets by it. Lemma for partim2 38190 and petlem 38195 via disjdmqseq 38188. (Contributed by Peter Mazsa, 16-Sep-2021.)

Assertion

Ref	Expression
disjdmqs	⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))

Proof of Theorem disjdmqs

Step	Hyp	Ref	Expression
1		disjdmqsss 38185	. 2 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) ⊆ (dom ≀ 𝑅 / ≀ 𝑅))
2		disjdmqscossss 38186	. 2 ⊢ ( Disj 𝑅 → (dom ≀ 𝑅 / ≀ 𝑅) ⊆ (dom 𝑅 / 𝑅))
3	1, 2	eqssd 3994	1 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))

Syntax hints:   → wi 4   = wceq 1533  dom cdm 5669   / cqs 8704   ≀ ccoss 37556   Disj wdisjALTV 37590

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-qs 8711  df-coss 37794  df-cnvrefrel 37910  df-disjALTV 38088

This theorem is referenced by:  disjdmqseq  38188