Theorem resss 4737

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 4450	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss1 3220	. 2 |- ( A i^i ( B X. _V ) ) C_ A
3	1, 2	eqsstri 3056	1 |- ( A |` B ) C_ A

Syntax hints:   _Vcvv 2619   i^i cin 2998   C_ wss 2999   X. cxp 4436   |` cres 4440

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-res 4450

This theorem is referenced by:  relssres  4750  resexg  4752  iss  4758  cocnvres  4955  relresfld  4960  relcoi1  4962  funres  5055  funres11  5086  funcnvres  5087  2elresin  5125  fssres  5186  foimacnv  5271  tposss  6011  dftpos4  6028  smores  6057  smores2  6059  caserel  6776