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Theorem unidmqs 36041
Description: The range of a relation is equal to the union of the domain quotient. (Contributed by Peter Mazsa, 13-Oct-2018.)
Assertion
Ref Expression
unidmqs (𝑅𝑉 → (Rel 𝑅 (dom 𝑅 / 𝑅) = ran 𝑅))

Proof of Theorem unidmqs
StepHypRef Expression
1 resexg 5868 . . . 4 (𝑅𝑉 → (𝑅 ↾ dom 𝑅) ∈ V)
2 rnresequniqs 35742 . . . 4 ((𝑅 ↾ dom 𝑅) ∈ V → ran (𝑅 ↾ dom 𝑅) = (dom 𝑅 / 𝑅))
31, 2syl 17 . . 3 (𝑅𝑉 → ran (𝑅 ↾ dom 𝑅) = (dom 𝑅 / 𝑅))
4 resdm 5867 . . . 4 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
54rneqd 5776 . . 3 (Rel 𝑅 → ran (𝑅 ↾ dom 𝑅) = ran 𝑅)
63, 5sylan9req 2857 . 2 ((𝑅𝑉 ∧ Rel 𝑅) → (dom 𝑅 / 𝑅) = ran 𝑅)
76ex 416 1 (𝑅𝑉 → (Rel 𝑅 (dom 𝑅 / 𝑅) = ran 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  Vcvv 3444   cuni 4803  dom cdm 5523  ran crn 5524  cres 5525  Rel wrel 5528   / cqs 8275
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ec 8278  df-qs 8282
This theorem is referenced by:  unidmqseq  36042
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