MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmresexg Unicode version

Theorem dmresexg 4978
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 4976 . 2  |-  dom  ( A  |`  B )  =  ( B  i^i  dom  A )
2 inex1g 4157 . 2  |-  ( B  e.  V  ->  ( B  i^i  dom  A )  e.  _V )
31, 2syl5eqel 2367 1  |-  ( B  e.  V  ->  dom  ( A  |`  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684   _Vcvv 2788    i^i cin 3151   dom cdm 4689    |` cres 4691
This theorem is referenced by:  resfunexg  5737  resfunexgALT  5738
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-dm 4699  df-res 4701
  Copyright terms: Public domain W3C validator