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Theorem imaundi 5000
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 4876 . . . 4  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
21rneqi 4812 . . 3  |-  ran  (  A  |`  ( B  u.  C ) )  =  ran  ( ( A  |`  B )  u.  ( A  |`  C ) )
3 rnun 4996 . . 3  |-  ran  (
( A  |`  B )  u.  ( A  |`  C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
42, 3eqtri 2273 . 2  |-  ran  (  A  |`  ( B  u.  C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
5 df-ima 4601 . 2  |-  ( A
" ( B  u.  C ) )  =  ran  (  A  |`  ( B  u.  C
) )
6 df-ima 4601 . . 3  |-  ( A
" B )  =  ran  (  A  |`  B )
7 df-ima 4601 . . 3  |-  ( A
" C )  =  ran  (  A  |`  C )
86, 7uneq12i 3237 . 2  |-  ( ( A " B )  u.  ( A " C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
94, 5, 83eqtr4i 2283 1  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    u. cun 3076   ran crn 4581    |` cres 4582   "cima 4583
This theorem is referenced by:  fnimapr  5435  domunfican  7014  fiint  7018  fodomfi  7020  marypha1lem  7070  dprd2da  15112  dmdprdsplit2lem  15115  uniioombllem3  18772  mbfimaicc  18820  plyeq0  19425  eupath2lem3  23074
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601
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