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

Theorem imain 5328
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 5327 . . 3  |-  ( Fun  `' F  ->  ( F
" ( A  \ 
( A  \  B
) ) )  =  ( ( F " A )  \  ( F " ( A  \  B ) ) ) )
2 imadif 5327 . . . 4  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
32difeq2d 3294 . . 3  |-  ( Fun  `' F  ->  ( ( F " A ) 
\  ( F "
( A  \  B
) ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) ) )
41, 3eqtrd 2315 . 2  |-  ( Fun  `' F  ->  ( F
" ( A  \ 
( A  \  B
) ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) ) )
5 dfin4 3409 . . 3  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
65imaeq2i 5010 . 2  |-  ( F
" ( A  i^i  B ) )  =  ( F " ( A 
\  ( A  \  B ) ) )
7 dfin4 3409 . 2  |-  ( ( F " A )  i^i  ( F " B ) )  =  ( ( F " A )  \  (
( F " A
)  \  ( F " B ) ) )
84, 6, 73eqtr4g 2340 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 1623    \ cdif 3149    i^i cin 3151   `'ccnv 4688   "cima 4692   Fun wfun 5249
This theorem is referenced by:  inpreima  5652  rnelfmlem  17647  fmfnfmlem3  17651  ballotlemfrc  23085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257
  Copyright terms: Public domain W3C validator