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Theorem ssres 5211
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 3637 . 2  |-  ( A 
C_  B  ->  ( A  i^i  ( C  X.  _V ) )  C_  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 4925 . 2  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
3 df-res 4925 . 2  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
41, 2, 33sstr4g 3458 1  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   _Vcvv 3034    i^i cin 3388    C_ wss 3389    X. cxp 4911    |` cres 4915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-v 3036  df-in 3396  df-ss 3403  df-res 4925
This theorem is referenced by:  imass1  5283  marypha1lem  7808  sspg  25758  ssps  25760  sspn  25766
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