Theorem resexg 4668

Description: The restriction of a set is a set. (Contributed by NM, 28-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
resexg	|- ( A e. V -> ( A |` B ) e. _V )

Proof of Theorem resexg

Step	Hyp	Ref	Expression
1		resss 4653	. 2 |- ( A |` B ) C_ A
2		ssexg 3917	. 2 |- ( ( ( A |` B ) C_ A /\ A e. V ) -> ( A |` B ) e. _V )
3	1, 2	mpan 414	1 |- ( A e. V -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 1433   _Vcvv 2601   C_ wss 2973   |` cres 4365

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-res 4375

This theorem is referenced by:  resex  4669  offres  5782  climres  10142