Theorem resexg 5045

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 5029	. 2 |- ( A |` B ) C_ A
2		ssexg 4223	. 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 2799   C_ wss 3197   |` cres 4721

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 4202

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 2801  df-in 3203  df-ss 3210  df-res 4731

This theorem is referenced by:  resex  5046  offres  6280  resixp  6880  seqf1oglem2  10742  climres  11814  setsvalg  13062  setsex  13064  setsslid  13083  gsumsplit1r  13431  znval  14600  znle  14601  znbaslemnn  14603  znleval  14617