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

Theorem imaundi 5284
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 5160 . . . 4  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
21rneqi 5096 . . 3  |-  ran  ( A  |`  ( B  u.  C ) )  =  ran  ( ( A  |`  B )  u.  ( A  |`  C ) )
3 rnun 5280 . . 3  |-  ran  (
( A  |`  B )  u.  ( A  |`  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
42, 3eqtri 2456 . 2  |-  ran  ( A  |`  ( B  u.  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
5 df-ima 4891 . 2  |-  ( A
" ( B  u.  C ) )  =  ran  ( A  |`  ( B  u.  C
) )
6 df-ima 4891 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
7 df-ima 4891 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
86, 7uneq12i 3499 . 2  |-  ( ( A " B )  u.  ( A " C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
94, 5, 83eqtr4i 2466 1  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3318   ran crn 4879    |` cres 4880   "cima 4881
This theorem is referenced by:  fnimapr  5787  domunfican  7379  fiint  7383  fodomfi  7385  marypha1lem  7438  dprd2da  15600  dmdprdsplit2lem  15603  uniioombllem3  19477  mbfimaicc  19525  plyeq0  20130  eupath2lem3  21701  mbfposadd  26254  itg2addnclem2  26257  ftc1anclem1  26280  ftc1anclem5  26284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
  Copyright terms: Public domain W3C validator