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Theorem fssres 5429
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 5258 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 fnssres 5367 . . . . 5  |-  ( ( F  Fn  A  /\  C  C_  A )  -> 
( F  |`  C )  Fn  C )
3 resss 4966 . . . . . . 7  |-  ( F  |`  C )  C_  F
4 rnss 4892 . . . . . . 7  |-  ( ( F  |`  C )  C_  F  ->  ran  ( F  |`  C )  C_  ran  F )
53, 4ax-mp 5 . . . . . 6  |-  ran  ( F  |`  C )  C_  ran  F
6 sstr 3187 . . . . . 6  |-  ( ( ran  ( F  |`  C )  C_  ran  F  /\  ran  F  C_  B )  ->  ran  ( F  |`  C ) 
C_  B )
75, 6mpan 424 . . . . 5  |-  ( ran 
F  C_  B  ->  ran  ( F  |`  C ) 
C_  B )
82, 7anim12i 338 . . . 4  |-  ( ( ( F  Fn  A  /\  C  C_  A )  /\  ran  F  C_  B )  ->  (
( F  |`  C )  Fn  C  /\  ran  ( F  |`  C ) 
C_  B ) )
98an32s 568 . . 3  |-  ( ( ( F  Fn  A  /\  ran  F  C_  B
)  /\  C  C_  A
)  ->  ( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B ) )
101, 9sylanb 284 . 2  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B
) )
11 df-f 5258 . 2  |-  ( ( F  |`  C ) : C --> B  <->  ( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B ) )
1210, 11sylibr 134 1  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    C_ wss 3153   ran crn 4660    |` cres 4661    Fn wfn 5249   -->wf 5250
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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-fun 5256  df-fn 5257  df-f 5258
This theorem is referenced by:  fssresd  5430  fssres2  5431  fresin  5432  f1ssres  5468  feqresmpt  5611  f2ndf  6279  elmapssres  6727  pmresg  6730  finomni  7199  fseq1p1m1  10160  seqf1oglem2  10591  wrdred1  10956  resmhm  13059  resghm  13330  hmeores  14483  limcdifap  14816  012of  15491  2o01f  15492
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