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Theorem fssres2 5409
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 5408 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( F  |`  A )  |`  C ) : C --> B )
2 resabs1 4984 . . . 4  |-  ( C 
C_  A  ->  (
( F  |`  A )  |`  C )  =  ( F  |`  C )
)
32feq1d 5379 . . 3  |-  ( C 
C_  A  ->  (
( ( F  |`  A )  |`  C ) : C --> B  <->  ( F  |`  C ) : C --> B ) )
43adantl 452 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( ( F  |`  A )  |`  C ) : C --> B  <->  ( F  |`  C ) : C --> B ) )
51, 4mpbid 201 1  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( F  |`  C ) : C --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    C_ wss 3152    |` cres 4691   -->wf 5251
This theorem is referenced by:  efcvx  19825  filnetlem4  26330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-fun 5257  df-fn 5258  df-f 5259
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