Theorem disjdmqseq 39275

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 39276 (which is the closest theorem to the former prter2 39373). Lemma for partim2 39277 and petlem 39282. (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 39274	. 2 ⊢ ( Disj 𝑅 → (dom 𝑅 / 𝑅) = (dom ≀ 𝑅 / ≀ 𝑅))
2	1	eqeq1d 2741	1 ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 207   = wceq 1547  dom cdm 5618   / cqs 8632   ≀ ccoss 38550   Disj wdisjALTV 38586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rmo 3344  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ec 8635  df-qs 8639  df-coss 38868  df-cnvrefrel 38974  df-disjALTV 39157

This theorem is referenced by:  eldisjn0el  39276  partim2  39277  petlem  39282