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Theorem imaundir 5094
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 4702 . . 3  |-  ( ( A  u.  B )
" C )  =  ran  ( ( A  u.  B )  |`  C )
2 resundir 4970 . . . 4  |-  ( ( A  u.  B )  |`  C )  =  ( ( A  |`  C )  u.  ( B  |`  C ) )
32rneqi 4905 . . 3  |-  ran  (
( A  u.  B
)  |`  C )  =  ran  ( ( A  |`  C )  u.  ( B  |`  C ) )
4 rnun 5089 . . 3  |-  ran  (
( A  |`  C )  u.  ( B  |`  C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
51, 3, 43eqtri 2307 . 2  |-  ( ( A  u.  B )
" C )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
6 df-ima 4702 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
7 df-ima 4702 . . 3  |-  ( B
" C )  =  ran  ( B  |`  C )
86, 7uneq12i 3327 . 2  |-  ( ( A " C )  u.  ( B " C ) )  =  ( ran  ( A  |`  C )  u.  ran  ( B  |`  C ) )
95, 8eqtr4i 2306 1  |-  ( ( A  u.  B )
" C )  =  ( ( A " C )  u.  ( B " C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3150   ran crn 4690    |` cres 4691   "cima 4692
This theorem is referenced by:  fvun  5589  fpwwe2lem13  8264  gsumzaddlem  15203  mbfres2  19000  funsnfsup  26762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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