Theorem resss 4984

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 4688	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss1 3393	. 2 |- ( A i^i ( B X. _V ) ) C_ A
3	1, 2	eqsstri 3225	1 |- ( A |` B ) C_ A

Syntax hints:   _Vcvv 2772   i^i cin 3165   C_ wss 3166   X. cxp 4674   |` cres 4678

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-res 4688

This theorem is referenced by:  relssres  4998  resexg  5000  iss  5006  cocnvres  5208  relresfld  5213  relcoi1  5215  funres  5313  funres11  5347  funcnvres  5348  2elresin  5388  fssres  5453  foimacnv  5542  tposss  6334  dftpos4  6351  smores  6380  smores2  6382  caserel  7191  txss12  14771  txbasval  14772