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Theorem imain 5344
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 5343 . . 3  |-  ( Fun  `' F  ->  ( F
" ( A  \ 
( A  \  B
) ) )  =  ( ( F " A )  \  ( F " ( A  \  B ) ) ) )
2 imadif 5343 . . . 4  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
32difeq2d 3307 . . 3  |-  ( Fun  `' F  ->  ( ( F " A ) 
\  ( F "
( A  \  B
) ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) ) )
41, 3eqtrd 2328 . 2  |-  ( Fun  `' F  ->  ( F
" ( A  \ 
( A  \  B
) ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) ) )
5 dfin4 3422 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65imaeq2i 5026 . 2  |-  ( F
" ( A  i^i  B ) )  =  ( F " ( A 
\  ( A  \  B ) ) )
7 dfin4 3422 . 2  |-  ( ( F " A )  i^i  ( F " B ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) )
84, 6, 73eqtr4g 2353 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 1632    \ cdif 3162    i^i cin 3164   `'ccnv 4704   "cima 4708   Fun wfun 5265
This theorem is referenced by:  inpreima  5668  rnelfmlem  17663  fmfnfmlem3  17667  ballotlemfrc  23101  spthispth  28359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273
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