Theorem resexg 4924

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 4908	. 2 |- ( A |` B ) C_ A
2		ssexg 4121	. 2 |- ( ( ( A |` B ) C_ A /\ A e. V ) -> ( A |` B ) e. _V )
3	1, 2	mpan 421	1 |- ( A e. V -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2136   _Vcvv 2726   C_ wss 3116   |` cres 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-sep 4100

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-res 4616

This theorem is referenced by:  resex  4925  offres  6103  resixp  6699  climres  11244  setsvalg  12424  setsex  12426  setsslid  12444