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Theorem ssres 5299
Description: Subclass theorem for restriction. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
ssres  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )

Proof of Theorem ssres
StepHypRef Expression
1 ssrin 3723 . 2  |-  ( A 
C_  B  ->  ( A  i^i  ( C  X.  _V ) )  C_  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 5011 . 2  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
3 df-res 5011 . 2  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
41, 2, 33sstr4g 3545 1  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   _Vcvv 3113    i^i cin 3475    C_ wss 3476    X. cxp 4997    |` cres 5001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-in 3483  df-ss 3490  df-res 5011
This theorem is referenced by:  imass1  5371  marypha1lem  7894  sspg  25414  ssps  25416  sspn  25422
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