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

Theorem dmresv 5132
Description: The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
Assertion
Ref Expression
dmresv  |-  dom  ( A  |`  _V )  =  dom  A

Proof of Theorem dmresv
StepHypRef Expression
1 dmres 4976 . 2  |-  dom  ( A  |`  _V )  =  ( _V  i^i  dom  A )
2 incom 3361 . 2  |-  ( _V 
i^i  dom  A )  =  ( dom  A  i^i  _V )
3 inv1 3481 . 2  |-  ( dom 
A  i^i  _V )  =  dom  A
41, 2, 33eqtri 2307 1  |-  dom  ( A  |`  _V )  =  dom  A
Colors of variables: wff set class
Syntax hints:    = wceq 1623   _Vcvv 2788    i^i cin 3151   dom cdm 4689    |` cres 4691
This theorem is referenced by:  fidomdm  7138  dmct  23342
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