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Theorem ssres2 5173
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 4984 . . 3  |-  ( A 
C_  B  ->  ( A  X.  _V )  C_  ( B  X.  _V )
)
2 sslin 3567 . . 3  |-  ( ( A  X.  _V )  C_  ( B  X.  _V )  ->  ( C  i^i  ( A  X.  _V )
)  C_  ( C  i^i  ( B  X.  _V ) ) )
31, 2syl 16 . 2  |-  ( A 
C_  B  ->  ( C  i^i  ( A  X.  _V ) )  C_  ( C  i^i  ( B  X.  _V ) ) )
4 df-res 4890 . 2  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
5 df-res 4890 . 2  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
63, 4, 53sstr4g 3389 1  |-  ( A 
C_  B  ->  ( C  |`  A )  C_  ( C  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2956    i^i cin 3319    C_ wss 3320    X. cxp 4876    |` cres 4880
This theorem is referenced by:  imass2  5240  1stcof  6374  2ndcof  6375  tfrlem15  6653  gsum2d  15546  txkgen  17684  funpsstri  25389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-in 3327  df-ss 3334  df-opab 4267  df-xp 4884  df-res 4890
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