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Theorem imaundir 6101
Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
Assertion
Ref Expression
imaundir ((𝐴𝐵) “ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem imaundir
StepHypRef Expression
1 df-ima 5644 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
2 resundir 5950 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
32rneqi 5890 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∪ (𝐵𝐶))
4 rnun 6096 . . 3 ran ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
51, 3, 43eqtri 2768 . 2 ((𝐴𝐵) “ 𝐶) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
6 df-ima 5644 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 5644 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7uneq12i 4119 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
95, 8eqtr4i 2767 1 ((𝐴𝐵) “ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  cun 3906  ran crn 5632  cres 5633  cima 5634
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-12 2171  ax-ext 2707
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-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644
This theorem is referenced by:  fvun  6928  suppun  8111  fsuppun  9320  fpwwe2lem12  10574  ustuqtop1  23577  mbfres2  24993  imadifxp  31405  eulerpartlemt  32840  bj-projun  35432  bj-funun  35690  poimirlem3  36048  poimirlem15  36060  brtrclfv2  41941  frege131d  41978  unhe1  41999  frege110  42187  frege133  42210  aacllem  47180
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