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Theorem resin 5852
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 5851 . . . 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 5838 . . . 4  |-  ( ( F  |`  ( A  \  B ) ) : ( A  \  B
)
-1-1-onto-> ( C  \  D )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( C  \  D ) )
31, 2syl 17 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( C  \  D ) )
4 resdif 5851 . . 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 1309 . 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 3713 . . . 4  |-  ( C  i^i  D )  =  ( C  \  ( C  \  D ) )
7 f1oeq3 5824 . . . 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 5 . . 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 3713 . . . 4  |-  ( A  i^i  B )  =  ( A  \  ( A  \  B ) )
10 f1oeq2 5823 . . . 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 5 . . 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 5121 . . . 4  |-  ( F  |`  ( A  i^i  B
) )  =  ( F  |`  ( A  \  ( A  \  B
) ) )
13 f1oeq1 5822 . . . 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 5 . . 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 275 . 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 199 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437    \ cdif 3433    i^i cin 3435   `'ccnv 4852    |` cres 4855   Fun wfun 5595   -onto->wfo 5599   -1-1-onto->wf1o 5600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608
This theorem is referenced by: (None)
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