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

Theorem ssres 4840
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 3296 . 2  |-  ( A 
C_  B  ->  ( A  i^i  ( C  X.  _V ) )  C_  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 4546 . 2  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
3 df-res 4546 . 2  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
41, 2, 33sstr4g 3135 1  |-  ( A 
C_  B  ->  ( A  |`  C )  C_  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2681    i^i cin 3065    C_ wss 3066    X. cxp 4532    |` cres 4536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-res 4546
This theorem is referenced by:  imass1  4909
  Copyright terms: Public domain W3C validator