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Theorem dmresexg 5198
Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
dmresexg  |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )

Proof of Theorem dmresexg
StepHypRef Expression
1 dmres 5196 . 2  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 inex1g 4375 . 2  |-  ( B  e.  V  ->  ( B  i^i  dom  A )  e.  _V )
31, 2syl5eqel 2526 1  |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1727   _Vcvv 2962    i^i cin 3305   dom cdm 4907    |` cres 4909
This theorem is referenced by:  resfunexg  5986  resfunexgALT  5987
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 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-xp 4913  df-dm 4917  df-res 4919
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