Theorem resexg 5051

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 5035	. 2 |- ( A |` B ) C_ A
2		ssexg 4226	. 2 |- ( ( ( A |` B ) C_ A /\ A e. V ) -> ( A |` B ) e. _V )
3	1, 2	mpan 424	1 |- ( A e. V -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2200   _Vcvv 2800   C_ wss 3198   |` cres 4725

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-sep 4205

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-in 3204  df-ss 3211  df-res 4735

This theorem is referenced by:  resex  5052  offres  6292  resixp  6897  seqf1oglem2  10772  climres  11854  setsvalg  13102  setsex  13104  setsslid  13123  gsumsplit1r  13471  znval  14640  znle  14641  znbaslemnn  14643  znleval  14657