Theorem resss 4635

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 4357	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		inss1 3157	. 2 |- ( A i^i ( B X. _V ) ) C_ A
3	1, 2	eqsstri 2975	1 |- ( A |` B ) C_ A

Syntax hints:   _Vcvv 2557   i^i cin 2916   C_ wss 2917   X. cxp 4343   |` cres 4347

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-res 4357

This theorem is referenced by:  relssres  4648  resexg  4650  iss  4654  relresfld  4847  relcoi1  4849  funres  4941  funres11  4971  funcnvres  4972  2elresin  5010  fssres  5066  foimacnv  5144  tposss  5861  dftpos4  5878  smores  5907  smores2  5909