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Theorem dmqseqim 36117
 Description: If the domain quotient of a relation is equal to the class 𝐴, then the range of the relation is the union of the class. (Contributed by Peter Mazsa, 29-Dec-2021.)
Assertion
Ref Expression
dmqseqim (𝑅𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = 𝐴)))

Proof of Theorem dmqseqim
StepHypRef Expression
1 unieq 4814 . . 3 ((dom 𝑅 / 𝑅) = 𝐴 (dom 𝑅 / 𝑅) = 𝐴)
2 unidmqseq 36116 . . . 4 (𝑅𝑉 → (Rel 𝑅 → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))
32imp 410 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴))
41, 3syl5ib 247 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = 𝐴))
54ex 416 1 (𝑅𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∪ cuni 4803  dom cdm 5522  ran crn 5523  Rel wrel 5527   / cqs 8286 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7451 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3722  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4246  df-if 4428  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-xp 5528  df-rel 5529  df-cnv 5530  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-ec 8289  df-qs 8293 This theorem is referenced by:  dmqseqim2  36118
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