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Theorem imaundir 5350
Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
Assertion
Ref Expression
imaundir  |-  ( ( A  u.  B )
" C )  =  ( ( A " C )  u.  ( B " C ) )

Proof of Theorem imaundir
StepHypRef Expression
1 df-ima 4953 . . 3  |-  ( ( A  u.  B )
" C )  =  ran  ( ( A  u.  B )  |`  C )
2 resundir 5225 . . . 4  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
32rneqi 5166 . . 3  |-  ran  (
( A  u.  B
)  |`  C )  =  ran  ( ( A  |`  C )  u.  ( B  |`  C ) )
4 rnun 5345 . . 3  |-  ran  (
( A  |`  C )  u.  ( B  |`  C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
51, 3, 43eqtri 2484 . 2  |-  ( ( A  u.  B )
" C )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
6 df-ima 4953 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4953 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7uneq12i 3608 . 2  |-  ( ( A " C )  u.  ( B " C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
95, 8eqtr4i 2483 1  |-  ( ( A  u.  B )
" C )  =  ( ( A " C )  u.  ( B " C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    u. cun 3426   ran crn 4941    |` cres 4942   "cima 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953
This theorem is referenced by:  fvun  5862  suppun  6811  fsuppun  7742  fpwwe2lem13  8912  gsumzaddlemOLD  16516  funsnfsupOLD  17779  ustuqtop1  19934  mbfres2  21241  imadifxp  26075  eulerpartlemt  26890  bj-projun  32789
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