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Theorem fresin 5604
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 3553 . . 3  |-  ( A  i^i  X )  C_  A
2 fssres 5602 . . 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 5151 . . . 4  |-  ( ( F  |`  A )  |`  X )  =  ( F  |`  ( A  i^i  X ) )
5 ffn 5583 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
6 fnresdm 5546 . . . . . 6  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
75, 6syl 16 . . . . 5  |-  ( F : A --> B  -> 
( F  |`  A )  =  F )
87reseq1d 5137 . . . 4  |-  ( F : A --> B  -> 
( ( F  |`  A )  |`  X )  =  ( F  |`  X ) )
94, 8syl5eqr 2481 . . 3  |-  ( F : A --> B  -> 
( F  |`  ( A  i^i  X ) )  =  ( F  |`  X ) )
109feq1d 5572 . 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 1652    i^i cin 3311    C_ wss 3312    |` cres 4872    Fn wfn 5441   -->wf 5442
This theorem is referenced by:  o1res  12346  limcresi  19764  dvreslem  19788  dvres2lem  19789  noreson  25607  mbfresfi  26243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-fun 5448  df-fn 5449  df-f 5450
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