Theorem dmep 5948

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.

Step	Hyp	Ref	Expression
1		eqv 3498	. 2 ⊢ (dom E = V ↔ ∀𝑥 𝑥 ∈ dom E )
2		el 5457	. . . 4 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5602	. . . . 5 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1846	. . . 4 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 231	. . 3 ⊢ ∃𝑦 𝑥 E 𝑦
6		vex 3492	. . . 4 ⊢ 𝑥 ∈ V
7	6	eldm 5925	. . 3 ⊢ (𝑥 ∈ dom E ↔ ∃𝑦 𝑥 E 𝑦)
8	5, 7	mpbir 231	. 2 ⊢ 𝑥 ∈ dom E
9	1, 8	mpgbir 1797	1 ⊢ dom E = V

Syntax hints:   = wceq 1537  ∃wex 1777   ∈ wcel 2108  Vcvv 3488   class class class wbr 5166   E cep 5598  dom cdm 5700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-eprel 5599  df-dm 5710

This theorem is referenced by: (None)