Theorem disjdmqseq 39977

Description: If a relation is disjoint, its domain quotient is equal to a class if and only if the domain quotient of the cosets by it is equal to the class. General version of eldisjn0el 39978 (which is the closest theorem to the former prter2 40075). Lemma for partim2 39979 and petlem 39984. (Contributed by Peter Mazsa, 16-Sep-2021.)

Assertion

Ref	Expression
disjdmqseq	⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Proof of Theorem disjdmqseq

Step	Hyp	Ref	Expression
1		disjdmqs 39976	. 2 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))
2	1	eqeq1d 2762	1 ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569  dom cdm 5651   / cqs 8724   ≀ ccoss 39252   Disj wdisjALTV 39288

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8727  df-qs 8731  df-coss 39570  df-cnvrefrel 39676  df-disjALTV 39859

This theorem is used by:  eldisjn0el  39978  partim2  39979  petlem  39984