Theorem domep 32294

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 2059	. . . 4 ⊢ 𝑥 = 𝑥
2		el 5083	. . . . 5 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5271	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1892	. . . . 5 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 223	. . . 4 ⊢ ∃𝑦 𝑥 E 𝑦
6	1, 5	2th 256	. . 3 ⊢ (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦)
7	6	abbii 2908	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8		df-v 3400	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9		df-dm 5367	. 2 ⊢ dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
10	7, 8, 9	3eqtr4ri 2813	1 ⊢ dom E = V

Syntax hints:   = wceq 1601  ∃wex 1823  {cab 2763  Vcvv 3398   class class class wbr 4888   E cep 5267  dom cdm 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-eprel 5268  df-dm 5367

This theorem is referenced by: (None)