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Theorem fresin 5554
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 3506 . . 3  |-  ( A  i^i  X )  C_  A
2 fssres 5552 . . 3  |-  ( ( F : A --> B  /\  ( A  i^i  X ) 
C_  A )  -> 
( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B )
31, 2mpan2 653 . 2  |-  ( F : A --> B  -> 
( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B )
4 resres 5101 . . . 4  |-  ( ( F  |`  A )  |`  X )  =  ( F  |`  ( A  i^i  X ) )
5 ffn 5533 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
6 fnresdm 5496 . . . . . 6  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
75, 6syl 16 . . . . 5  |-  ( F : A --> B  -> 
( F  |`  A )  =  F )
87reseq1d 5087 . . . 4  |-  ( F : A --> B  -> 
( ( F  |`  A )  |`  X )  =  ( F  |`  X ) )
94, 8syl5eqr 2435 . . 3  |-  ( F : A --> B  -> 
( F  |`  ( A  i^i  X ) )  =  ( F  |`  X ) )
109feq1d 5522 . 2  |-  ( F : A --> B  -> 
( ( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B  <-> 
( F  |`  X ) : ( A  i^i  X ) --> B ) )
113, 10mpbid 202 1  |-  ( F : A --> B  -> 
( F  |`  X ) : ( A  i^i  X ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    i^i cin 3264    C_ wss 3265    |` cres 4822    Fn wfn 5391   -->wf 5392
This theorem is referenced by:  o1res  12283  limcresi  19641  dvreslem  19665  dvres2lem  19666  noreson  25340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-fun 5398  df-fn 5399  df-f 5400
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