Theorem resss 5062

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 4761	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss1 3441	. 2 |- ( A i^i ( B X. _V ) ) C_ A
3	1, 2	eqsstri 3270	1 |- ( A |` B ) C_ A

Syntax hints:   _Vcvv 2813   i^i cin 3210   C_ wss 3211   X. cxp 4747   |` cres 4751

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-in 3217  df-ss 3224  df-res 4761

This theorem is referenced by:  relssres  5076  resexg  5078  iss  5084  cocnvres  5287  relresfld  5292  relcoi1  5294  funres  5393  funres11  5428  funcnvres  5429  2elresin  5469  fssres  5540  foimacnv  5632  tposss  6477  dftpos4  6494  smores  6523  smores2  6525  caserel  7378  txss12  15131  txbasval  15132  issubgr2  16253  subgrprop2  16255  uhgrspansubgr  16272