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Theorem imain 5488
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 5487 . . 3  |-  ( Fun  `' F  ->  ( F
" ( A  \ 
( A  \  B
) ) )  =  ( ( F " A )  \  ( F " ( A  \  B ) ) ) )
2 imadif 5487 . . . 4  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
32difeq2d 3425 . . 3  |-  ( Fun  `' F  ->  ( ( F " A ) 
\  ( F "
( A  \  B
) ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) ) )
41, 3eqtrd 2436 . 2  |-  ( Fun  `' F  ->  ( F
" ( A  \ 
( A  \  B
) ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) ) )
5 dfin4 3541 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65imaeq2i 5160 . 2  |-  ( F
" ( A  i^i  B ) )  =  ( F " ( A 
\  ( A  \  B ) ) )
7 dfin4 3541 . 2  |-  ( ( F " A )  i^i  ( F " B ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) )
84, 6, 73eqtr4g 2461 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 1649    \ cdif 3277    i^i cin 3279   `'ccnv 4836   "cima 4840   Fun wfun 5407
This theorem is referenced by:  inpreima  5816  rnelfmlem  17937  fmfnfmlem3  17941  spthispth  21526  ballotlemfrc  24737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-fun 5415
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