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Theorem dmep 5924
Description: The domain of the membership relation is the universal class. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof shortened by BJ, 26-Dec-2023.)
Assertion
Ref Expression
dmep dom E = V

Proof of Theorem dmep
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqv 3484 . 2 (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E )
2 el 5438 . . . 4 𝑦 𝑥𝑦
3 epel 5584 . . . . 5 (𝑥 E 𝑦𝑥𝑦)
43exbii 1851 . . . 4 (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥𝑦)
52, 4mpbir 230 . . 3 𝑦 𝑥 E 𝑦
6 vex 3479 . . . 4 𝑥 ∈ V
76eldm 5901 . . 3 (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦)
85, 7mpbir 230 . 2 𝑥 ∈ dom E
91, 8mpgbir 1802 1 dom E = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475   class class class wbr 5149   E cep 5580  dom cdm 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-dm 5687
This theorem is referenced by: (None)
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