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Theorem unidmqseq 37167
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 37166 . . . 4 (𝑅𝑉 → (Rel 𝑅 (dom 𝑅 / 𝑅) = ran 𝑅))
21imp 408 . . 3 ((𝑅𝑉 ∧ Rel 𝑅) → (dom 𝑅 / 𝑅) = ran 𝑅)
32eqeq1d 2735 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → ( (dom 𝑅 / 𝑅) = 𝐴 ↔ ran 𝑅 = 𝐴))
43ex 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 4869  dom cdm 5637  ran crn 5638  Rel wrel 5642   / cqs 8653
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 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ec 8656  df-qs 8660
This theorem is referenced by:  dmqseqim  37168
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