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Theorem imaundi 5350
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 5225 . . . 4  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
21rneqi 5167 . . 3  |-  ran  ( A  |`  ( B  u.  C ) )  =  ran  ( ( A  |`  B )  u.  ( A  |`  C ) )
3 rnun 5346 . . 3  |-  ran  (
( A  |`  B )  u.  ( A  |`  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
42, 3eqtri 2480 . 2  |-  ran  ( A  |`  ( B  u.  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
5 df-ima 4954 . 2  |-  ( A
" ( B  u.  C ) )  =  ran  ( A  |`  ( B  u.  C
) )
6 df-ima 4954 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
7 df-ima 4954 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
86, 7uneq12i 3609 . 2  |-  ( ( A " B )  u.  ( A " C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
94, 5, 83eqtr4i 2490 1  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    u. cun 3427   ran crn 4942    |` cres 4943   "cima 4944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-br 4394  df-opab 4452  df-xp 4947  df-cnv 4949  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954
This theorem is referenced by:  fnimapr  5857  domunfican  7688  fiint  7692  fodomfi  7694  marypha1lem  7787  dprd2da  16655  dmdprdsplit2lem  16658  uniioombllem3  21191  mbfimaicc  21237  plyeq0  21805  eupath2lem3  23745  ffsrn  26173  mbfposadd  28580  itg2addnclem2  28585  ftc1anclem1  28608  ftc1anclem5  28612
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