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Theorem ssres2 4998
Description: Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ssres2  |-  ( A 
C_  B  ->  ( C  |`  A )  C_  ( C  |`  B ) )

Proof of Theorem ssres2
StepHypRef Expression
1 xpss1 4811 . . 3  |-  ( A 
C_  B  ->  ( A  X.  _V )  C_  ( B  X.  _V )
)
2 sslin 3408 . . 3  |-  ( ( A  X.  _V )  C_  ( B  X.  _V )  ->  ( C  i^i  ( A  X.  _V )
)  C_  ( C  i^i  ( B  X.  _V ) ) )
31, 2syl 15 . 2  |-  ( A 
C_  B  ->  ( C  i^i  ( A  X.  _V ) )  C_  ( C  i^i  ( B  X.  _V ) ) )
4 df-res 4717 . 2  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
5 df-res 4717 . 2  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
63, 4, 53sstr4g 3232 1  |-  ( A 
C_  B  ->  ( C  |`  A )  C_  ( C  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2801    i^i cin 3164    C_ wss 3165    X. cxp 4703    |` cres 4707
This theorem is referenced by:  imass2  5065  1stcof  6163  2ndcof  6164  tfrlem15  6424  gsum2d  15239  txkgen  17362  funpsstri  24192  int2pre  25340
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2803  df-in 3172  df-ss 3179  df-opab 4094  df-xp 4711  df-res 4717
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