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Theorem domep 28830
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 1740 . . . 4  |-  x  =  x
2 el 4629 . . . . 5  |-  E. y  x  e.  y
3 epel 4794 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
43exbii 1644 . . . . 5  |-  ( E. y  x  _E  y  <->  E. y  x  e.  y )
52, 4mpbir 209 . . . 4  |-  E. y  x  _E  y
61, 52th 239 . . 3  |-  ( x  =  x  <->  E. y  x  _E  y )
76abbii 2601 . 2  |-  { x  |  x  =  x }  =  { x  |  E. y  x  _E  y }
8 df-v 3115 . 2  |-  _V  =  { x  |  x  =  x }
9 df-dm 5009 . 2  |-  dom  _E  =  { x  |  E. y  x  _E  y }
107, 8, 93eqtr4ri 2507 1  |-  dom  _E  =  _V
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   E.wex 1596   {cab 2452   _Vcvv 3113   class class class wbr 4447    _E cep 4789   dom cdm 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-eprel 4791  df-dm 5009
This theorem is referenced by: (None)
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