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

Theorem resss 4927
Description: A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.)
Assertion
Ref Expression
resss  |-  ( A  |`  B )  C_  A

Proof of Theorem resss
StepHypRef Expression
1 df-res 4635 . 2  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 inss1 3355 . 2  |-  ( A  i^i  ( B  X.  _V ) )  C_  A
31, 2eqsstri 3187 1  |-  ( A  |`  B )  C_  A
Colors of variables: wff set class
Syntax hints:   _Vcvv 2737    i^i cin 3128    C_ wss 3129    X. cxp 4621    |` cres 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-res 4635
This theorem is referenced by:  relssres  4941  resexg  4943  iss  4949  cocnvres  5149  relresfld  5154  relcoi1  5156  funres  5253  funres11  5284  funcnvres  5285  2elresin  5323  fssres  5387  foimacnv  5475  tposss  6241  dftpos4  6258  smores  6287  smores2  6289  caserel  7080  txss12  13430  txbasval  13431
  Copyright terms: Public domain W3C validator