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Theorem resin 5511
Description: The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
Assertion
Ref Expression
resin  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) -1-1-onto-> ( C  i^i  D ) )

Proof of Theorem resin
StepHypRef Expression
1 resdif 5510 . . . 4  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -1-1-onto-> ( C 
\  D ) )
2 f1ofo 5495 . . . 4  |-  ( ( F  |`  ( A  \  B ) ) : ( A  \  B
)
-1-1-onto-> ( C  \  D )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( C  \  D ) )
31, 2syl 15 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( C  \  D ) )
4 resdif 5510 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  ( A 
\  B ) ) : ( A  \  B ) -onto-> ( C 
\  D ) )  ->  ( F  |`  ( A  \  ( A  \  B ) ) ) : ( A 
\  ( A  \  B ) ) -1-1-onto-> ( C 
\  ( C  \  D ) ) )
53, 4syld3an3 1227 . 2  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  \  ( A  \  B ) ) ) : ( A 
\  ( A  \  B ) ) -1-1-onto-> ( C 
\  ( C  \  D ) ) )
6 dfin4 3422 . . . 4  |-  ( C  i^i  D )  =  ( C  \  ( C  \  D ) )
7 f1oeq3 5481 . . . 4  |-  ( ( C  i^i  D )  =  ( C  \ 
( C  \  D
) )  ->  (
( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) -1-1-onto-> ( C  i^i  D
)  <->  ( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) -1-1-onto-> ( C 
\  ( C  \  D ) ) ) )
86, 7ax-mp 8 . . 3  |-  ( ( F  |`  ( A  i^i  B ) ) : ( A  i^i  B
)
-1-1-onto-> ( C  i^i  D )  <-> 
( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) -1-1-onto-> ( C  \  ( C  \  D ) ) )
9 dfin4 3422 . . . 4  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
10 f1oeq2 5480 . . . 4  |-  ( ( A  i^i  B )  =  ( A  \ 
( A  \  B
) )  ->  (
( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) -1-1-onto-> ( C  \  ( C  \  D ) )  <-> 
( F  |`  ( A  i^i  B ) ) : ( A  \ 
( A  \  B
) ) -1-1-onto-> ( C  \  ( C  \  D ) ) ) )
119, 10ax-mp 8 . . 3  |-  ( ( F  |`  ( A  i^i  B ) ) : ( A  i^i  B
)
-1-1-onto-> ( C  \  ( C  \  D ) )  <-> 
( F  |`  ( A  i^i  B ) ) : ( A  \ 
( A  \  B
) ) -1-1-onto-> ( C  \  ( C  \  D ) ) )
129reseq2i 4968 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  =  ( F  |`  ( A  \  ( A  \  B
) ) )
13 f1oeq1 5479 . . . 4  |-  ( ( F  |`  ( A  i^i  B ) )  =  ( F  |`  ( A  \  ( A  \  B ) ) )  ->  ( ( F  |`  ( A  i^i  B
) ) : ( A  \  ( A 
\  B ) ) -1-1-onto-> ( C  \  ( C 
\  D ) )  <-> 
( F  |`  ( A  \  ( A  \  B ) ) ) : ( A  \ 
( A  \  B
) ) -1-1-onto-> ( C  \  ( C  \  D ) ) ) )
1412, 13ax-mp 8 . . 3  |-  ( ( F  |`  ( A  i^i  B ) ) : ( A  \  ( A  \  B ) ) -1-1-onto-> ( C  \  ( C 
\  D ) )  <-> 
( F  |`  ( A  \  ( A  \  B ) ) ) : ( A  \ 
( A  \  B
) ) -1-1-onto-> ( C  \  ( C  \  D ) ) )
158, 11, 143bitrri 263 . 2  |-  ( ( F  |`  ( A  \  ( A  \  B
) ) ) : ( A  \  ( A  \  B ) ) -1-1-onto-> ( C  \  ( C 
\  D ) )  <-> 
( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) -1-1-onto-> ( C  i^i  D
) )
165, 15sylib 188 1  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) -1-1-onto-> ( C  i^i  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1632    \ cdif 3162    i^i cin 3164   `'ccnv 4704    |` cres 4707   Fun wfun 5265   -onto->wfo 5269   -1-1-onto->wf1o 5270
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  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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