Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  domep Unicode version

Theorem domep 24149
Description: The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
Assertion
Ref Expression
domep  |-  dom  _E  =  _V

Proof of Theorem domep
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 equid 1644 . . . 4  |-  x  =  x
2 el 4192 . . . . 5  |-  E. y  x  e.  y
3 epel 4308 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
43exbii 1569 . . . . 5  |-  ( E. y  x  _E  y  <->  E. y  x  e.  y )
52, 4mpbir 200 . . . 4  |-  E. y  x  _E  y
61, 52th 230 . . 3  |-  ( x  =  x  <->  E. y  x  _E  y )
76abbii 2395 . 2  |-  { x  |  x  =  x }  =  { x  |  E. y  x  _E  y }
8 df-v 2790 . 2  |-  _V  =  { x  |  x  =  x }
9 df-dm 4699 . 2  |-  dom  _E  =  { x  |  E. y  x  _E  y }
107, 8, 93eqtr4ri 2314 1  |-  dom  _E  =  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   class class class wbr 4023    _E cep 4303   dom cdm 4689
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-13 1686  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-pow 4188  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-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  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-eprel 4305  df-dm 4699
  Copyright terms: Public domain W3C validator