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Theorem dmep 5934
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 3490 . 2 (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E )
2 el 5442 . . . 4 𝑦 𝑥𝑦
3 epel 5587 . . . . 5 (𝑥 E 𝑦𝑥𝑦)
43exbii 1848 . . . 4 (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥𝑦)
52, 4mpbir 231 . . 3 𝑦 𝑥 E 𝑦
6 vex 3484 . . . 4 𝑥 ∈ V
76eldm 5911 . . 3 (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦)
85, 7mpbir 231 . 2 𝑥 ∈ dom E
91, 8mpgbir 1799 1 dom E = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480   class class class wbr 5143   E cep 5583  dom cdm 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-dm 5695
This theorem is referenced by: (None)
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