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Theorem fresin 5760
Description: An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
fresin  |-  ( F : A --> B  -> 
( F  |`  X ) : ( A  i^i  X ) --> B )

Proof of Theorem fresin
StepHypRef Expression
1 inss1 3723 . . 3  |-  ( A  i^i  X )  C_  A
2 fssres 5757 . . 3  |-  ( ( F : A --> B  /\  ( A  i^i  X ) 
C_  A )  -> 
( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B )
31, 2mpan2 671 . 2  |-  ( F : A --> B  -> 
( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B )
4 resres 5292 . . . 4  |-  ( ( F  |`  A )  |`  X )  =  ( F  |`  ( A  i^i  X ) )
5 ffn 5737 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
6 fnresdm 5696 . . . . . 6  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
75, 6syl 16 . . . . 5  |-  ( F : A --> B  -> 
( F  |`  A )  =  F )
87reseq1d 5278 . . . 4  |-  ( F : A --> B  -> 
( ( F  |`  A )  |`  X )  =  ( F  |`  X ) )
94, 8syl5eqr 2522 . . 3  |-  ( F : A --> B  -> 
( F  |`  ( A  i^i  X ) )  =  ( F  |`  X ) )
109feq1d 5723 . 2  |-  ( F : A --> B  -> 
( ( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B  <-> 
( F  |`  X ) : ( A  i^i  X ) --> B ) )
113, 10mpbid 210 1  |-  ( F : A --> B  -> 
( F  |`  X ) : ( A  i^i  X ) --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    i^i cin 3480    C_ wss 3481    |` cres 5007    Fn wfn 5589   -->wf 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-fun 5596  df-fn 5597  df-f 5598
This theorem is referenced by:  o1res  13362  limcresi  22155  dvreslem  22179  dvres2lem  22180  noreson  29354  mbfresfi  29995  limcresiooub  31513  limcresioolb  31514  limcleqr  31515  limclner  31522  mbfres2cn  31605  fouriersw  31861
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