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Theorem fssres2 5347
Description: Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
Assertion
Ref Expression
fssres2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )

Proof of Theorem fssres2
StepHypRef Expression
1 fssres 5345 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( F  |`  A )  |`  C ) : C --> B )
2 resabs1 4895 . . . 4  |-  ( C 
C_  A  ->  (
( F  |`  A )  |`  C )  =  ( F  |`  C )
)
32feq1d 5306 . . 3  |-  ( C 
C_  A  ->  (
( ( F  |`  A )  |`  C ) : C --> B  <->  ( F  |`  C ) : C --> B ) )
43adantl 275 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( ( F  |`  A )  |`  C ) : C --> B  <->  ( F  |`  C ) : C --> B ) )
51, 4mpbid 146 1  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    C_ wss 3102    |` cres 4588   -->wf 5166
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-fun 5172  df-fn 5173  df-f 5174
This theorem is referenced by:  frecsuclem  6353
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