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Theorem imaundi 5109
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 4985 . . . 4  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
21rneqi 4921 . . 3  |-  ran  ( A  |`  ( B  u.  C ) )  =  ran  ( ( A  |`  B )  u.  ( A  |`  C ) )
3 rnun 5105 . . 3  |-  ran  (
( A  |`  B )  u.  ( A  |`  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
42, 3eqtri 2316 . 2  |-  ran  ( A  |`  ( B  u.  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
5 df-ima 4718 . 2  |-  ( A
" ( B  u.  C ) )  =  ran  ( A  |`  ( B  u.  C
) )
6 df-ima 4718 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
7 df-ima 4718 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
86, 7uneq12i 3340 . 2  |-  ( ( A " B )  u.  ( A " C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
94, 5, 83eqtr4i 2326 1  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163   ran crn 4706    |` cres 4707   "cima 4708
This theorem is referenced by:  fnimapr  5599  domunfican  7145  fiint  7149  fodomfi  7151  marypha1lem  7202  dprd2da  15293  dmdprdsplit2lem  15296  uniioombllem3  18956  mbfimaicc  19004  plyeq0  19609  eupath2lem3  23918  itg2addnclem2  25004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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