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Theorem fssres 5298
Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
fssres  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )

Proof of Theorem fssres
StepHypRef Expression
1 df-f 5127 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 fnssres 5236 . . . . 5  |-  ( ( F  Fn  A  /\  C  C_  A )  -> 
( F  |`  C )  Fn  C )
3 resss 4843 . . . . . . 7  |-  ( F  |`  C )  C_  F
4 rnss 4769 . . . . . . 7  |-  ( ( F  |`  C )  C_  F  ->  ran  ( F  |`  C )  C_  ran  F )
53, 4ax-mp 5 . . . . . 6  |-  ran  ( F  |`  C )  C_  ran  F
6 sstr 3105 . . . . . 6  |-  ( ( ran  ( F  |`  C )  C_  ran  F  /\  ran  F  C_  B )  ->  ran  ( F  |`  C ) 
C_  B )
75, 6mpan 420 . . . . 5  |-  ( ran 
F  C_  B  ->  ran  ( F  |`  C ) 
C_  B )
82, 7anim12i 336 . . . 4  |-  ( ( ( F  Fn  A  /\  C  C_  A )  /\  ran  F  C_  B )  ->  (
( F  |`  C )  Fn  C  /\  ran  ( F  |`  C ) 
C_  B ) )
98an32s 557 . . 3  |-  ( ( ( F  Fn  A  /\  ran  F  C_  B
)  /\  C  C_  A
)  ->  ( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B ) )
101, 9sylanb 282 . 2  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B
) )
11 df-f 5127 . 2  |-  ( ( F  |`  C ) : C --> B  <->  ( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B ) )
1210, 11sylibr 133 1  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    C_ wss 3071   ran crn 4540    |` cres 4541    Fn wfn 5118   -->wf 5119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-fun 5125  df-fn 5126  df-f 5127
This theorem is referenced by:  fssresd  5299  fssres2  5300  fresin  5301  f1ssres  5337  feqresmpt  5475  f2ndf  6123  elmapssres  6567  pmresg  6570  finomni  7012  fseq1p1m1  9881  hmeores  12494  limcdifap  12810  isomninnlem  13255
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