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Theorem resss 4813
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 4521 . 2  |-  ( A  |`  B )  =  ( A  i^i  ( B  X.  _V ) )
2 inss1 3266 . 2  |-  ( A  i^i  ( B  X.  _V ) )  C_  A
31, 2eqsstri 3099 1  |-  ( A  |`  B )  C_  A
Colors of variables: wff set class
Syntax hints:   _Vcvv 2660    i^i cin 3040    C_ wss 3041    X. cxp 4507    |` cres 4511
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-res 4521
This theorem is referenced by:  relssres  4827  resexg  4829  iss  4835  cocnvres  5033  relresfld  5038  relcoi1  5040  funres  5134  funres11  5165  funcnvres  5166  2elresin  5204  fssres  5268  foimacnv  5353  tposss  6111  dftpos4  6128  smores  6157  smores2  6159  caserel  6940  txss12  12362  txbasval  12363
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