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Theorem imaundir 6172
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 5701 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
2 resundir 6014 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
32rneqi 5950 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∪ (𝐵𝐶))
4 rnun 6167 . . 3 ran ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
51, 3, 43eqtri 2766 . 2 ((𝐴𝐵) “ 𝐶) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
6 df-ima 5701 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 5701 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7uneq12i 4175 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
95, 8eqtr4i 2765 1 ((𝐴𝐵) “ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  cun 3960  ran crn 5689  cres 5690  cima 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701
This theorem is referenced by:  fvun  6998  suppun  8207  fsuppun  9424  fpwwe2lem12  10679  ustuqtop1  24265  mbfres2  25693  imadifxp  32620  suppun2  32698  eulerpartlemt  34352  bj-projun  36976  bj-funun  37234  poimirlem3  37609  poimirlem15  37621  brtrclfv2  43716  frege131d  43753  unhe1  43774  frege110  43962  frege133  43985  aacllem  49031
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