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

Theorem imaundi 5248
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 5118 . . . 4  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
21rneqi 5061 . . 3  |-  ran  ( A  |`  ( B  u.  C ) )  =  ran  ( ( A  |`  B )  u.  ( A  |`  C ) )
3 rnun 5244 . . 3  |-  ran  (
( A  |`  B )  u.  ( A  |`  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
42, 3eqtri 2473 . 2  |-  ran  ( A  |`  ( B  u.  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
5 df-ima 4847 . 2  |-  ( A
" ( B  u.  C ) )  =  ran  ( A  |`  ( B  u.  C
) )
6 df-ima 4847 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
7 df-ima 4847 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
86, 7uneq12i 3586 . 2  |-  ( ( A " B )  u.  ( A " C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
94, 5, 83eqtr4i 2483 1  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1444    u. cun 3402   ran crn 4835    |` cres 4836   "cima 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847
This theorem is referenced by:  fnimapr  5929  domunfican  7844  fiint  7848  fodomfi  7850  marypha1lem  7947  dprd2da  17675  dmdprdsplit2lem  17678  uniioombllem3  22543  mbfimaicc  22589  plyeq0  23165  eupath2lem3  25707  ffsrn  28314  imadifss  31928  poimirlem1  31941  poimirlem2  31942  poimirlem3  31943  poimirlem4  31944  poimirlem6  31946  poimirlem7  31947  poimirlem11  31951  poimirlem12  31952  poimirlem15  31955  poimirlem16  31956  poimirlem17  31957  poimirlem19  31959  poimirlem20  31960  poimirlem23  31963  poimirlem24  31964  poimirlem25  31965  poimirlem29  31969  poimirlem31  31971  mbfposadd  31988  itg2addnclem2  31994  ftc1anclem1  32017  ftc1anclem5  32021  brtrclfv2  36319  frege77d  36338  frege109d  36349  frege131d  36356  dffrege76  36535  icccncfext  37765
  Copyright terms: Public domain W3C validator