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Theorem imaundir 5761
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 5323 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
2 resundir 5620 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
32rneqi 5553 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∪ (𝐵𝐶))
4 rnun 5756 . . 3 ran ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
51, 3, 43eqtri 2823 . 2 ((𝐴𝐵) “ 𝐶) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
6 df-ima 5323 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 5323 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7uneq12i 3961 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
95, 8eqtr4i 2822 1 ((𝐴𝐵) “ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  cun 3765  ran crn 5311  cres 5312  cima 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-cnv 5318  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323
This theorem is referenced by:  fvun  6491  suppun  7550  fsuppun  8534  fpwwe2lem13  9750  ustuqtop1  22370  mbfres2  23750  imadifxp  29923  eulerpartlemt  30941  bj-projun  33466  poimirlem3  33893  poimirlem15  33905  brtrclfv2  38790  frege131d  38827  unhe1  38849  frege110  39037  frege133  39060  aacllem  43337
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