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Theorem dmep 5758
 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 3449 . 2 (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E )
2 el 5236 . . . 4 𝑦 𝑥𝑦
3 epel 5434 . . . . 5 (𝑥 E 𝑦𝑥𝑦)
43exbii 1849 . . . 4 (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥𝑦)
52, 4mpbir 234 . . 3 𝑦 𝑥 E 𝑦
6 vex 3444 . . . 4 𝑥 ∈ V
76eldm 5734 . . 3 (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦)
85, 7mpbir 234 . 2 𝑥 ∈ dom E
91, 8mpgbir 1801 1 dom E = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   class class class wbr 5031   E cep 5430  dom cdm 5520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-eprel 5431  df-dm 5530 This theorem is referenced by: (None)
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