ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssres2 Unicode version

Theorem ssres2 4697
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 4506 . . 3  |-  ( A 
C_  B  ->  ( A  X.  _V )  C_  ( B  X.  _V )
)
2 sslin 3210 . . 3  |-  ( ( A  X.  _V )  C_  ( B  X.  _V )  ->  ( C  i^i  ( A  X.  _V )
)  C_  ( C  i^i  ( B  X.  _V ) ) )
31, 2syl 14 . 2  |-  ( A 
C_  B  ->  ( C  i^i  ( A  X.  _V ) )  C_  ( C  i^i  ( B  X.  _V ) ) )
4 df-res 4413 . 2  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
5 df-res 4413 . 2  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
63, 4, 53sstr4g 3051 1  |-  ( A 
C_  B  ->  ( C  |`  A )  C_  ( C  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2612    i^i cin 2983    C_ wss 2984    X. cxp 4399    |` cres 4403
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997  df-opab 3866  df-xp 4407  df-res 4413
This theorem is referenced by:  imass2  4763  resasplitss  5138  1stcof  5869  2ndcof  5870
  Copyright terms: Public domain W3C validator