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Theorem ssres 5285
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 3705 . 2  |-  ( A 
C_  B  ->  ( A  i^i  ( C  X.  _V ) )  C_  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 4997 . 2  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
3 df-res 4997 . 2  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
41, 2, 33sstr4g 3527 1  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   _Vcvv 3093    i^i cin 3457    C_ wss 3458    X. cxp 4983    |` cres 4987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-v 3095  df-in 3465  df-ss 3472  df-res 4997
This theorem is referenced by:  imass1  5357  marypha1lem  7891  sspg  25506  ssps  25508  sspn  25514
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