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Theorem imaundir 4858
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 4465 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
2 resundir 4740 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
32rneqi 4676 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∪ (𝐵𝐶))
4 rnun 4853 . . 3 ran ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
51, 3, 43eqtri 2113 . 2 ((𝐴𝐵) “ 𝐶) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
6 df-ima 4465 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 4465 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7uneq12i 3153 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
95, 8eqtr4i 2112 1 ((𝐴𝐵) “ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1290  cun 2998  ran crn 4453  cres 4454  cima 4455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-cnv 4460  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465
This theorem is referenced by:  fvun1  5383
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