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Theorem resexg 4994
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 4979 . 2  |-  ( A  |`  B )  C_  A
2 ssexg 4160 . 2  |-  ( ( ( A  |`  B ) 
C_  A  /\  A  e.  V )  ->  ( A  |`  B )  e. 
_V )
31, 2mpan 651 1  |-  ( A  e.  V  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788    C_ wss 3152    |` cres 4691
This theorem is referenced by:  resex  4995  offres  6092  resixp  6851  climres  12049  setsvalg  13171  setsid  13187  gsumval3  15191  gsum2d  15223  qtopres  17389  tsmspropd  17814  ulmss  19774  hhssva  21836  hhsssm  21837  hhshsslem1  21844  umgrares  23876  exidres  26568  exidresid  26569  fvtresfn  26763  lmhmlnmsplit  27185  pwssplit4  27191  usgrares  28115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-res 4701
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