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Theorem fssres2 5423
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 5421 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( F  |`  A )  |`  C ) : C --> B )
2 resabs1 4965 . . . 4  |-  ( C 
C_  A  ->  (
( F  |`  A )  |`  C )  =  ( F  |`  C )
)
32feq1d 5382 . . 3  |-  ( C 
C_  A  ->  (
( ( F  |`  A )  |`  C ) : C --> B  <->  ( F  |`  C ) : C --> B ) )
43adantl 277 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( ( F  |`  A )  |`  C ) : C --> B  <->  ( F  |`  C ) : C --> B ) )
51, 4mpbid 147 1  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    C_ wss 3153    |` cres 4657   -->wf 5242
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 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-fun 5248  df-fn 5249  df-f 5250
This theorem is referenced by:  frecsuclem  6450
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