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Theorem imain 5531
Description: The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
Assertion
Ref Expression
imain  |-  ( Fun  `' F  ->  ( F
" ( A  i^i  B ) )  =  ( ( F " A
)  i^i  ( F " B ) ) )

Proof of Theorem imain
StepHypRef Expression
1 imadif 5530 . . 3  |-  ( Fun  `' F  ->  ( F
" ( A  \ 
( A  \  B
) ) )  =  ( ( F " A )  \  ( F " ( A  \  B ) ) ) )
2 imadif 5530 . . . 4  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
32difeq2d 3467 . . 3  |-  ( Fun  `' F  ->  ( ( F " A ) 
\  ( F "
( A  \  B
) ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) ) )
41, 3eqtrd 2470 . 2  |-  ( Fun  `' F  ->  ( F
" ( A  \ 
( A  \  B
) ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) ) )
5 dfin4 3583 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65imaeq2i 5203 . 2  |-  ( F
" ( A  i^i  B ) )  =  ( F " ( A 
\  ( A  \  B ) ) )
7 dfin4 3583 . 2  |-  ( ( F " A )  i^i  ( F " B ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) )
84, 6, 73eqtr4g 2495 1  |-  ( Fun  `' F  ->  ( F
" ( A  i^i  B ) )  =  ( ( F " A
)  i^i  ( F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    \ cdif 3319    i^i cin 3321   `'ccnv 4879   "cima 4883   Fun wfun 5450
This theorem is referenced by:  inpreima  5859  rnelfmlem  17986  fmfnfmlem3  17990  spthispth  21575  ballotlemfrc  24786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-fun 5458
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