Theorem resss 5082

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 4781	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss1 3451	. 2 |- ( A i^i ( B X. _V ) ) C_ A
3	1, 2	eqsstri 3280	1 |- ( A |` B ) C_ A

Syntax hints:   _Vcvv 2821   i^i cin 3219   C_ wss 3220   X. cxp 4767   |` cres 4771

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-res 4781

This theorem is referenced by:  relssres  5096  resexg  5098  iss  5104  cocnvres  5307  relresfld  5312  relcoi1  5314  funres  5413  funres11  5448  funcnvres  5449  2elresin  5489  fssres  5560  foimacnv  5652  tposss  6507  dftpos4  6524  smores  6553  smores2  6555  caserel  7417  txss12  15290  txbasval  15291  issubgr2  16413  subgrprop2  16415  uhgrspansubgr  16432