Theorem resss 4887

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 4595	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss1 3327	. 2 |- ( A i^i ( B X. _V ) ) C_ A
3	1, 2	eqsstri 3160	1 |- ( A |` B ) C_ A

Syntax hints:   _Vcvv 2712   i^i cin 3101   C_ wss 3102   X. cxp 4581   |` cres 4585

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-res 4595

This theorem is referenced by:  relssres  4901  resexg  4903  iss  4909  cocnvres  5107  relresfld  5112  relcoi1  5114  funres  5208  funres11  5239  funcnvres  5240  2elresin  5278  fssres  5342  foimacnv  5429  tposss  6187  dftpos4  6204  smores  6233  smores2  6235  caserel  7021  txss12  12626  txbasval  12627