Theorem resss 4933

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

Step	Hyp	Ref	Expression
1		df-res 4640	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss1 3357	. 2 |- ( A i^i ( B X. _V ) ) C_ A
3	1, 2	eqsstri 3189	1 |- ( A |` B ) C_ A

Syntax hints:   _Vcvv 2739   i^i cin 3130   C_ wss 3131   X. cxp 4626   |` cres 4630

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 2741  df-in 3137  df-ss 3144  df-res 4640

This theorem is referenced by:  relssres  4947  resexg  4949  iss  4955  cocnvres  5155  relresfld  5160  relcoi1  5162  funres  5259  funres11  5290  funcnvres  5291  2elresin  5329  fssres  5393  foimacnv  5481  tposss  6249  dftpos4  6266  smores  6295  smores2  6297  caserel  7088  txss12  13851  txbasval  13852