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Theorem fresin 5413
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 3370 . . 3  |-  ( A  i^i  X )  C_  A
2 fssres 5410 . . 3  |-  ( ( F : A --> B  /\  ( A  i^i  X ) 
C_  A )  -> 
( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B )
31, 2mpan2 425 . 2  |-  ( F : A --> B  -> 
( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B )
4 resres 4937 . . . 4  |-  ( ( F  |`  A )  |`  X )  =  ( F  |`  ( A  i^i  X ) )
5 ffn 5384 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
6 fnresdm 5344 . . . . . 6  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
75, 6syl 14 . . . . 5  |-  ( F : A --> B  -> 
( F  |`  A )  =  F )
87reseq1d 4924 . . . 4  |-  ( F : A --> B  -> 
( ( F  |`  A )  |`  X )  =  ( F  |`  X ) )
94, 8eqtr3id 2236 . . 3  |-  ( F : A --> B  -> 
( F  |`  ( A  i^i  X ) )  =  ( F  |`  X ) )
109feq1d 5371 . 2  |-  ( F : A --> B  -> 
( ( F  |`  ( A  i^i  X ) ) : ( A  i^i  X ) --> B  <-> 
( F  |`  X ) : ( A  i^i  X ) --> B ) )
113, 10mpbid 147 1  |-  ( F : A --> B  -> 
( F  |`  X ) : ( A  i^i  X ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    i^i cin 3143    C_ wss 3144    |` cres 4646    Fn wfn 5230   -->wf 5231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-fun 5237  df-fn 5238  df-f 5239
This theorem is referenced by:  limcresi  14612
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