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Theorem ssres2 5137
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 4948 . . 3  |-  ( A 
C_  B  ->  ( A  X.  _V )  C_  ( B  X.  _V )
)
2 sslin 3649 . . 3  |-  ( ( A  X.  _V )  C_  ( B  X.  _V )  ->  ( C  i^i  ( A  X.  _V )
)  C_  ( C  i^i  ( B  X.  _V ) ) )
31, 2syl 17 . 2  |-  ( A 
C_  B  ->  ( C  i^i  ( A  X.  _V ) )  C_  ( C  i^i  ( B  X.  _V ) ) )
4 df-res 4851 . 2  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
5 df-res 4851 . 2  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
63, 4, 53sstr4g 3459 1  |-  ( A 
C_  B  ->  ( C  |`  A )  C_  ( C  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   _Vcvv 3031    i^i cin 3389    C_ wss 3390    X. cxp 4837    |` cres 4841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-in 3397  df-ss 3404  df-opab 4455  df-xp 4845  df-res 4851
This theorem is referenced by:  imass2  5210  1stcof  6840  2ndcof  6841  tfrlem15  7128  gsum2dlem2  17681  txkgen  20744  funpsstri  30477  resnonrel  36269  mptrcllem  36291  rtrclexi  36299  cnvrcl0  36303  relexpss1d  36368  relexp0a  36379
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