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Theorem domep 25420
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 1688 . . . 4  |-  x  =  x
2 el 4381 . . . . 5  |-  E. y  x  e.  y
3 epel 4497 . . . . . 6  |-  ( x  _E  y  <->  x  e.  y )
43exbii 1592 . . . . 5  |-  ( E. y  x  _E  y  <->  E. y  x  e.  y )
52, 4mpbir 201 . . . 4  |-  E. y  x  _E  y
61, 52th 231 . . 3  |-  ( x  =  x  <->  E. y  x  _E  y )
76abbii 2548 . 2  |-  { x  |  x  =  x }  =  { x  |  E. y  x  _E  y }
8 df-v 2958 . 2  |-  _V  =  { x  |  x  =  x }
9 df-dm 4888 . 2  |-  dom  _E  =  { x  |  E. y  x  _E  y }
107, 8, 93eqtr4ri 2467 1  |-  dom  _E  =  _V
Colors of variables: wff set class
Syntax hints:   E.wex 1550    = wceq 1652   {cab 2422   _Vcvv 2956   class class class wbr 4212    _E cep 4492   dom cdm 4878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-eprel 4494  df-dm 4888
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