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Theorem resss 4663
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 4383 . 2  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 inss1 3193 . 2  |-  ( A  i^i  ( B  X.  _V ) )  C_  A
31, 2eqsstri 3030 1  |-  ( A  |`  B )  C_  A
Colors of variables: wff set class
Syntax hints:   _Vcvv 2602    i^i cin 2973    C_ wss 2974    X. cxp 4369    |` cres 4373
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-res 4383
This theorem is referenced by:  relssres  4676  resexg  4678  iss  4684  relresfld  4877  relcoi1  4879  funres  4971  funres11  5002  funcnvres  5003  2elresin  5041  fssres  5097  foimacnv  5175  tposss  5895  dftpos4  5912  smores  5941  smores2  5943
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