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Theorem imaundi 5092
Description: Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
imaundi  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )

Proof of Theorem imaundi
StepHypRef Expression
1 resundi 4968 . . . 4  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
21rneqi 4904 . . 3  |-  ran  (  A  |`  ( B  u.  C ) )  =  ran  ( ( A  |`  B )  u.  ( A  |`  C ) )
3 rnun 5088 . . 3  |-  ran  (
( A  |`  B )  u.  ( A  |`  C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
42, 3eqtri 2304 . 2  |-  ran  (  A  |`  ( B  u.  C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
5 df-ima 4701 . 2  |-  ( A
" ( B  u.  C ) )  =  ran  (  A  |`  ( B  u.  C
) )
6 df-ima 4701 . . 3  |-  ( A
" B )  =  ran  (  A  |`  B )
7 df-ima 4701 . . 3  |-  ( A
" C )  =  ran  (  A  |`  C )
86, 7uneq12i 3328 . 2  |-  ( ( A " B )  u.  ( A " C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
94, 5, 83eqtr4i 2314 1  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    u. cun 3151   ran crn 4689    |` cres 4690   "cima 4691
This theorem is referenced by:  fnimapr  5545  domunfican  7125  fiint  7129  fodomfi  7131  marypha1lem  7182  dprd2da  15273  dmdprdsplit2lem  15276  uniioombllem3  18936  mbfimaicc  18984  plyeq0  19589  eupath2lem3  23310
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 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
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 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701
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