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Theorem imaundir 5410
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 5005 . . 3  |-  ( ( A  u.  B )
" C )  =  ran  ( ( A  u.  B )  |`  C )
2 resundir 5279 . . . 4  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
32rneqi 5220 . . 3  |-  ran  (
( A  u.  B
)  |`  C )  =  ran  ( ( A  |`  C )  u.  ( B  |`  C ) )
4 rnun 5405 . . 3  |-  ran  (
( A  |`  C )  u.  ( B  |`  C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
51, 3, 43eqtri 2493 . 2  |-  ( ( A  u.  B )
" C )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
6 df-ima 5005 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 5005 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7uneq12i 3649 . 2  |-  ( ( A " C )  u.  ( B " C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
95, 8eqtr4i 2492 1  |-  ( ( A  u.  B )
" C )  =  ( ( A " C )  u.  ( B " C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    u. cun 3467   ran crn 4993    |` cres 4994   "cima 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005
This theorem is referenced by:  fvun  5928  suppun  6910  fsuppun  7837  fpwwe2lem13  9009  gsumzaddlemOLD  16720  funsnfsupOLD  18020  ustuqtop1  20472  mbfres2  21780  imadifxp  26981  eulerpartlemt  27800  bj-projun  33508
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