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Theorem resin 5495
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 5494 . . . 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 5479 . . . 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 5494 . . 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 3409 . . . 4  |-  ( C  i^i  D )  =  ( C  \  ( C  \  D ) )
7 f1oeq3 5465 . . . 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 3409 . . . 4  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
10 f1oeq2 5464 . . . 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 4952 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  =  ( F  |`  ( A  \  ( A  \  B
) ) )
13 f1oeq1 5463 . . . 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 1623    \ cdif 3149    i^i cin 3151   `'ccnv 4688    |` cres 4691   Fun wfun 5249   -onto->wfo 5253   -1-1-onto->wf1o 5254
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  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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