Theorem resexg 4940

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 4924	. 2 |- ( A |` B ) C_ A
2		ssexg 4137	. 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 2146   _Vcvv 2735   C_ wss 3127   |` cres 4622

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-sep 4116

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-ss 3140  df-res 4632

This theorem is referenced by:  resex  4941  offres  6126  resixp  6723  climres  11279  setsvalg  12459  setsex  12461  setsslid  12479