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Theorem ssres2 4982
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 4795 . . 3  |-  ( A 
C_  B  ->  ( A  X.  _V )  C_  ( B  X.  _V )
)
2 sslin 3395 . . 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 4701 . 2  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
5 df-res 4701 . 2  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
63, 4, 53sstr4g 3219 1  |-  ( A 
C_  B  ->  ( C  |`  A )  C_  ( C  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2788    i^i cin 3151    C_ wss 3152    X. cxp 4687    |` cres 4691
This theorem is referenced by:  imass2  5049  1stcof  6147  2ndcof  6148  tfrlem15  6408  gsum2d  15223  txkgen  17346  funpsstri  24121  int2pre  25237
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-v 2790  df-in 3159  df-ss 3166  df-opab 4078  df-xp 4695  df-res 4701
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