Theorem domep 30942

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 1926	. . . 4 ⊢ 𝑥 = 𝑥
2		el 4773	. . . . 5 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 4952	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1764	. . . . 5 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 220	. . . 4 ⊢ ∃𝑦 𝑥 E 𝑦
6	1, 5	2th 253	. . 3 ⊢ (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦)
7	6	abbii 2726	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8		df-v 3175	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9		df-dm 5048	. 2 ⊢ dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
10	7, 8, 9	3eqtr4ri 2643	1 ⊢ dom E = V

Syntax hints:   = wceq 1475  ∃wex 1695  {cab 2596  Vcvv 3173   class class class wbr 4583   E cep 4947  dom cdm 5038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-eprel 4949  df-dm 5048

This theorem is referenced by: (None)