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Theorem resexg 4854
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
StepHypRef Expression
1 resss 4838 . 2  |-  ( A  |`  B )  C_  A
2 ssexg 4062 . 2  |-  ( ( ( A  |`  B ) 
C_  A  /\  A  e.  V )  ->  ( A  |`  B )  e. 
_V )
31, 2mpan 420 1  |-  ( A  e.  V  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480   _Vcvv 2681    C_ wss 3066    |` cres 4536
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-res 4546
This theorem is referenced by:  resex  4855  offres  6026  resixp  6620  climres  11065  setsvalg  11978  setsex  11980  setsslid  11998
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