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Theorem rnresequniqs 36783
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 36780 . 2 ((𝑅𝐴) ∈ 𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
2 df-ima 5646 . 2 (𝑅𝐴) = ran (𝑅𝐴)
31, 2eqtr2di 2793 1 ((𝑅𝐴) ∈ 𝑉 → ran (𝑅𝐴) = (𝐴 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106   cuni 4865  ran crn 5634  cres 5635  cima 5636   / cqs 8646
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ec 8649  df-qs 8653
This theorem is referenced by:  unidmqs  37106
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