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Theorem imaundir 5173
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 4781 . . 3  |-  ( ( A  u.  B )
" C )  =  ran  ( ( A  u.  B )  |`  C )
2 resundir 5049 . . . 4  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
32rneqi 4984 . . 3  |-  ran  (
( A  u.  B
)  |`  C )  =  ran  ( ( A  |`  C )  u.  ( B  |`  C ) )
4 rnun 5168 . . 3  |-  ran  (
( A  |`  C )  u.  ( B  |`  C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
51, 3, 43eqtri 2382 . 2  |-  ( ( A  u.  B )
" C )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
6 df-ima 4781 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4781 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7uneq12i 3403 . 2  |-  ( ( A " C )  u.  ( B " C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
95, 8eqtr4i 2381 1  |-  ( ( A  u.  B )
" C )  =  ( ( A " C )  u.  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1642    u. cun 3226   ran crn 4769    |` cres 4770   "cima 4771
This theorem is referenced by:  fvun  5669  fpwwe2lem13  8351  gsumzaddlem  15296  mbfres2  19098  imadifxp  23238  ustuqtop1  23545  funsnfsup  26085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-cnv 4776  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781
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