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Theorem fssres2 5613
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 5612 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( F  |`  A )  |`  C ) : C --> B )
2 resabs1 5177 . . . 4  |-  ( C 
C_  A  ->  (
( F  |`  A )  |`  C )  =  ( F  |`  C )
)
32feq1d 5582 . . 3  |-  ( C 
C_  A  ->  (
( ( F  |`  A )  |`  C ) : C --> B  <->  ( F  |`  C ) : C --> B ) )
43adantl 454 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( ( F  |`  A )  |`  C ) : C --> B  <->  ( F  |`  C ) : C --> B ) )
51, 4mpbid 203 1  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    C_ wss 3322    |` cres 4882   -->wf 5452
This theorem is referenced by:  efcvx  20367  filnetlem4  26412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-fun 5458  df-fn 5459  df-f 5460
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