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Theorem rnresequniqs 35707
 Description: The range of a restriction is equal to the union of the quotient set. (Contributed by Peter Mazsa, 19-May-2018.)
Assertion
Ref Expression
rnresequniqs ((𝑅𝐴) ∈ 𝑉 → ran (𝑅𝐴) = (𝐴 / 𝑅))

Proof of Theorem rnresequniqs
StepHypRef Expression
1 uniqsALTV 35704 . 2 ((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
2 df-ima 5545 . 2 (𝑅𝐴) = ran (𝑅𝐴)
31, 2syl6req 2874 1 ((𝑅𝐴) ∈ 𝑉 → ran (𝑅𝐴) = (𝐴 / 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2114  ∪ cuni 4813  ran crn 5533   ↾ cres 5534   “ cima 5535   / 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-ec 8278  df-qs 8282 This theorem is referenced by:  unidmqs  36006
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