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Theorem unidmqseq 38656
Description: The union of the domain quotient of a relation is equal to the class 𝐴 if and only if the range is equal to it as well. (Contributed by Peter Mazsa, 21-Apr-2019.) (Revised by Peter Mazsa, 28-Dec-2021.)
Assertion
Ref Expression
unidmqseq (𝑅𝑉 → (Rel 𝑅 → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))

Proof of Theorem unidmqseq
StepHypRef Expression
1 unidmqs 38655 . . . 4 (𝑅𝑉 → (Rel 𝑅 (dom 𝑅 / 𝑅) = ran 𝑅))
21imp 406 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → (dom 𝑅 / 𝑅) = ran 𝑅)
32eqeq1d 2739 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴))
43ex 412 1 (𝑅𝑉 → (Rel 𝑅 → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108   cuni 4907  dom cdm 5685  ran crn 5686  Rel wrel 5690   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751
This theorem is referenced by:  dmqseqim  38657
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