MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imaundi Unicode version

Theorem imaundi 5081
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 4957 . . . 4  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
21rneqi 4893 . . 3  |-  ran  (  A  |`  ( B  u.  C ) )  =  ran  ( ( A  |`  B )  u.  ( A  |`  C ) )
3 rnun 5077 . . 3  |-  ran  (
( A  |`  B )  u.  ( A  |`  C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
42, 3eqtri 2278 . 2  |-  ran  (  A  |`  ( B  u.  C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
5 df-ima 4682 . 2  |-  ( A
" ( B  u.  C ) )  =  ran  (  A  |`  ( B  u.  C
) )
6 df-ima 4682 . . 3  |-  ( A
" B )  =  ran  (  A  |`  B )
7 df-ima 4682 . . 3  |-  ( A
" C )  =  ran  (  A  |`  C )
86, 7uneq12i 3302 . 2  |-  ( ( A " B )  u.  ( A " C ) )  =  ( ran  (  A  |`  B )  u.  ran  (  A  |`  C ) )
94, 5, 83eqtr4i 2288 1  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1619    u. cun 3125   ran crn 4662    |` cres 4663   "cima 4664
This theorem is referenced by:  fnimapr  5517  domunfican  7097  fiint  7101  fodomfi  7103  marypha1lem  7154  dprd2da  15240  dmdprdsplit2lem  15243  uniioombllem3  18903  mbfimaicc  18951  plyeq0  19556  eupath2lem3  23276
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 1927  ax-ext 2239
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 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682
  Copyright terms: Public domain W3C validator