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Theorem fssres 5117
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 4956 . . 3  |-  ( F : A --> B  <->  ( F  Fn  A  /\  ran  F  C_  B ) )
2 fnssres 5063 . . . . 5  |-  ( ( F  Fn  A  /\  C  C_  A )  -> 
( F  |`  C )  Fn  C )
3 resss 4683 . . . . . . 7  |-  ( F  |`  C )  C_  F
4 rnss 4612 . . . . . . 7  |-  ( ( F  |`  C )  C_  F  ->  ran  ( F  |`  C )  C_  ran  F )
53, 4ax-mp 7 . . . . . 6  |-  ran  ( F  |`  C )  C_  ran  F
6 sstr 3016 . . . . . 6  |-  ( ( ran  ( F  |`  C )  C_  ran  F  /\  ran  F  C_  B )  ->  ran  ( F  |`  C ) 
C_  B )
75, 6mpan 415 . . . . 5  |-  ( ran 
F  C_  B  ->  ran  ( F  |`  C ) 
C_  B )
82, 7anim12i 331 . . . 4  |-  ( ( ( F  Fn  A  /\  C  C_  A )  /\  ran  F  C_  B )  ->  (
( F  |`  C )  Fn  C  /\  ran  ( F  |`  C ) 
C_  B ) )
98an32s 533 . . 3  |-  ( ( ( F  Fn  A  /\  ran  F  C_  B
)  /\  C  C_  A
)  ->  ( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B ) )
101, 9sylanb 278 . 2  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B
) )
11 df-f 4956 . 2  |-  ( ( F  |`  C ) : C --> B  <->  ( ( F  |`  C )  Fn  C  /\  ran  ( F  |`  C )  C_  B ) )
1210, 11sylibr 132 1  |-  ( ( F : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    C_ wss 2982   ran crn 4392    |` cres 4393    Fn wfn 4947   -->wf 4948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-fun 4954  df-fn 4955  df-f 4956
This theorem is referenced by:  fssres2  5118  fresin  5119  f1ssres  5150  feqresmpt  5279  f2ndf  5898  fseq1p1m1  9239
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