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Theorem imaundir 5288
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 4894 . . 3  |-  ( ( A  u.  B )
" C )  =  ran  ( ( A  u.  B )  |`  C )
2 resundir 5164 . . . 4  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
32rneqi 5099 . . 3  |-  ran  (
( A  u.  B
)  |`  C )  =  ran  ( ( A  |`  C )  u.  ( B  |`  C ) )
4 rnun 5283 . . 3  |-  ran  (
( A  |`  C )  u.  ( B  |`  C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
51, 3, 43eqtri 2462 . 2  |-  ( ( A  u.  B )
" C )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
6 df-ima 4894 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4894 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7uneq12i 3501 . 2  |-  ( ( A " C )  u.  ( B " C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
95, 8eqtr4i 2461 1  |-  ( ( A  u.  B )
" C )  =  ( ( A " C )  u.  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    u. cun 3320   ran crn 4882    |` cres 4883   "cima 4884
This theorem is referenced by:  fvun  5796  fpwwe2lem13  8522  gsumzaddlem  15531  ustuqtop1  18276  mbfres2  19540  imadifxp  24043  funsnfsup  26757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894
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