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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.
StepHypRef Expression
1 equid 2094 . . . 4 𝑥 = 𝑥
2 el 4996 . . . . 5 𝑦 𝑥𝑦
3 epel 5182 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
43exbii 1923 . . . . 5 (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥𝑦)
52, 4mpbir 221 . . . 4 𝑦 𝑥 E 𝑦
61, 52th 254 . . 3 (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦)
76abbii 2877 . 2 {𝑥𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8 df-v 3342 . 2 V = {𝑥𝑥 = 𝑥}
9 df-dm 5276 . 2 dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
107, 8, 93eqtr4ri 2793 1 dom E = V
Colors of variables: wff setvar class
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)
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