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Theorem domep 24220
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 1662 . . . 4  |-  x  =  x
2 el 4208 . . . . 5  |-  E. y  x  e.  y
3 epel 4324 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
43exbii 1572 . . . . 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 2408 . 2  |-  { x  |  x  =  x }  =  { x  |  E. y  x  _E  y }
8 df-v 2803 . 2  |-  _V  =  { x  |  x  =  x }
9 df-dm 4715 . 2  |-  dom  _E  =  { x  |  E. y  x  _E  y }
107, 8, 93eqtr4ri 2327 1  |-  dom  _E  =  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   _Vcvv 2801   class class class wbr 4039    _E cep 4319   dom cdm 4705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-eprel 4321  df-dm 4715
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