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Theorem fssres2 5425
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 5424 . 2  |-  ( ( ( F  |`  A ) : A --> B  /\  C  C_  A )  -> 
( ( F  |`  A )  |`  C ) : C --> B )
2 resabs1 5000 . . . 4  |-  ( C 
C_  A  ->  (
( F  |`  A )  |`  C )  =  ( F  |`  C )
)
32feq1d 5395 . . 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 3165    |` cres 4707   -->wf 5267
This theorem is referenced by:  efcvx  19841  filnetlem4  26433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-fun 5273  df-fn 5274  df-f 5275
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