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Theorem dmqseqim 37464
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 4918 . . 3 ((dom 𝑅 / 𝑅) = 𝐴 (dom 𝑅 / 𝑅) = 𝐴)
2 unidmqseq 37463 . . . 4 (𝑅𝑉 → (Rel 𝑅 → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴)))
32imp 408 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴))
41, 3imbitrid 243 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = 𝐴))
54ex 414 1 (𝑅𝑉 → (Rel 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 → ran 𝑅 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   cuni 4907  dom cdm 5675  ran crn 5676  Rel wrel 5680   / cqs 8698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ec 8701  df-qs 8705
This theorem is referenced by:  dmqseqim2  37465
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