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Theorem fresin 5595
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 3585 . . 3  |-  ( A  i^i  X )  C_  A
2 fssres 5593 . . 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 5138 . . . 4  |-  ( ( F  |`  A )  |`  X )  =  ( F  |`  ( A  i^i  X ) )
5 ffn 5574 . . . . . 6  |-  ( F : A --> B  ->  F  Fn  A )
6 fnresdm 5535 . . . . . 6  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
75, 6syl 16 . . . . 5  |-  ( F : A --> B  -> 
( F  |`  A )  =  F )
87reseq1d 5124 . . . 4  |-  ( F : A --> B  -> 
( ( F  |`  A )  |`  X )  =  ( F  |`  X ) )
94, 8syl5eqr 2489 . . 3  |-  ( F : A --> B  -> 
( F  |`  ( A  i^i  X ) )  =  ( F  |`  X ) )
109feq1d 5561 . 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 1369    i^i cin 3342    C_ wss 3343    |` cres 4857    Fn wfn 5428   -->wf 5429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-br 4308  df-opab 4366  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-fun 5435  df-fn 5436  df-f 5437
This theorem is referenced by:  o1res  13053  limcresi  21375  dvreslem  21399  dvres2lem  21400  noreson  27816  mbfresfi  28457
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