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Theorem resexg 4829
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 4813 . 2  |-  ( A  |`  B )  C_  A
2 ssexg 4037 . 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 1465   _Vcvv 2660    C_ wss 3041    |` cres 4511
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-res 4521
This theorem is referenced by:  resex  4830  offres  6001  resixp  6595  climres  11040  setsvalg  11916  setsex  11918  setsslid  11936
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