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Theorem imaundi 5409
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 5278 . . . 4  |-  ( A  |`  ( B  u.  C
) )  =  ( ( A  |`  B )  u.  ( A  |`  C ) )
21rneqi 5220 . . 3  |-  ran  ( A  |`  ( B  u.  C ) )  =  ran  ( ( A  |`  B )  u.  ( A  |`  C ) )
3 rnun 5405 . . 3  |-  ran  (
( A  |`  B )  u.  ( A  |`  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
42, 3eqtri 2489 . 2  |-  ran  ( A  |`  ( B  u.  C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
5 df-ima 5005 . 2  |-  ( A
" ( B  u.  C ) )  =  ran  ( A  |`  ( B  u.  C
) )
6 df-ima 5005 . . 3  |-  ( A
" B )  =  ran  ( A  |`  B )
7 df-ima 5005 . . 3  |-  ( A
" C )  =  ran  ( A  |`  C )
86, 7uneq12i 3649 . 2  |-  ( ( A " B )  u.  ( A " C ) )  =  ( ran  ( A  |`  B )  u.  ran  ( A  |`  C ) )
94, 5, 83eqtr4i 2499 1  |-  ( A
" ( B  u.  C ) )  =  ( ( A " B )  u.  ( A " C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    u. cun 3467   ran crn 4993    |` cres 4994   "cima 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005
This theorem is referenced by:  fnimapr  5922  domunfican  7782  fiint  7786  fodomfi  7788  marypha1lem  7882  dprd2da  16874  dmdprdsplit2lem  16877  uniioombllem3  21722  mbfimaicc  21768  plyeq0  22336  eupath2lem3  24641  ffsrn  27210  mbfposadd  29626  itg2addnclem2  29631  ftc1anclem1  29654  ftc1anclem5  29658  icccncfext  31181
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