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Theorem domep 31426
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 1936 . . . 4 𝑥 = 𝑥
2 el 4812 . . . . 5 𝑦 𝑥𝑦
3 epel 4993 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
43exbii 1771 . . . . 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 2736 . 2 {𝑥𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8 df-v 3191 . 2 V = {𝑥𝑥 = 𝑥}
9 df-dm 5089 . 2 dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
107, 8, 93eqtr4ri 2654 1 dom E = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wex 1701  {cab 2607  Vcvv 3189   class class class wbr 4618   E cep 4988  dom cdm 5079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-eprel 4990  df-dm 5089
This theorem is referenced by: (None)
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