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Theorem ssres 4726
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 3223 . 2  |-  ( A 
C_  B  ->  ( A  i^i  ( C  X.  _V ) )  C_  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 4440 . 2  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
3 df-res 4440 . 2  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
41, 2, 33sstr4g 3065 1  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2619    i^i cin 2996    C_ wss 2997    X. cxp 4426    |` cres 4430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-ss 3010  df-res 4440
This theorem is referenced by:  imass1  4794
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