Theorem domep 32003

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.

Step	Hyp	Ref	Expression
1		equid 2094	. . . 4 ⊢ 𝑥 = 𝑥
2		el 4996	. . . . 5 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5182	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1923	. . . . 5 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 221	. . . 4 ⊢ ∃𝑦 𝑥 E 𝑦
6	1, 5	2th 254	. . 3 ⊢ (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦)
7	6	abbii 2877	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8		df-v 3342	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9		df-dm 5276	. 2 ⊢ dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
10	7, 8, 9	3eqtr4ri 2793	1 ⊢ dom E = V

Syntax hints:   = wceq 1632  ∃wex 1853  {cab 2746  Vcvv 3340   class class class wbr 4804   E cep 5178  dom cdm 5266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-eprel 5179  df-dm 5276

This theorem is referenced by: (None)