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Theorem resexg 5188
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 5173 . 2  |-  ( A  |`  B )  C_  A
2 ssexg 4352 . 2  |-  ( ( ( A  |`  B ) 
C_  A  /\  A  e.  V )  ->  ( A  |`  B )  e. 
_V )
31, 2mpan 653 1  |-  ( A  e.  V  ->  ( A  |`  B )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726   _Vcvv 2958    C_ wss 3322    |` cres 4883
This theorem is referenced by:  resex  5189  offres  6322  resixp  7100  climres  12374  setsvalg  13497  setsid  13513  gsumval3  15519  gsum2d  15551  qtopres  17735  tsmspropd  18166  ulmss  20318  uhgrares  21348  umgrares  21364  usgrares  21394  usgrares1  21429  cusgrares  21486  redwlk  21611  hhssva  22764  hhsssm  22765  hhshsslem1  22772  exidres  26567  exidresid  26568  fvtresfn  26758  lmhmlnmsplit  27176  pwssplit4  27182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-ss 3336  df-res 4893
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