Theorem resss 4970

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 4675	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss1 3383	. 2 |- ( A i^i ( B X. _V ) ) C_ A
3	1, 2	eqsstri 3215	1 |- ( A |` B ) C_ A

Syntax hints:   _Vcvv 2763   i^i cin 3156   C_ wss 3157   X. cxp 4661   |` cres 4665

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-res 4675

This theorem is referenced by:  relssres  4984  resexg  4986  iss  4992  cocnvres  5194  relresfld  5199  relcoi1  5201  funres  5299  funres11  5330  funcnvres  5331  2elresin  5369  fssres  5433  foimacnv  5522  tposss  6304  dftpos4  6321  smores  6350  smores2  6352  caserel  7153  txss12  14502  txbasval  14503