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Theorem domepOLD 5759
 Description: Obsolete proof of dmep 5758 as of 26-Dec-2023. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
domepOLD dom E = V

Proof of Theorem domepOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 equid 2019 . . . 4 𝑥 = 𝑥
2 el 5236 . . . . 5 𝑦 𝑥𝑦
3 epel 5434 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
43exbii 1849 . . . . 5 (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥𝑦)
52, 4mpbir 234 . . . 4 𝑦 𝑥 E 𝑦
61, 52th 267 . . 3 (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦)
76abbii 2863 . 2 {𝑥𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8 df-v 3443 . 2 V = {𝑥𝑥 = 𝑥}
9 df-dm 5530 . 2 dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
107, 8, 93eqtr4ri 2832 1 dom E = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ∃wex 1781  {cab 2776  Vcvv 3441   class class class wbr 5031   E cep 5430  dom cdm 5520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-eprel 5431  df-dm 5530 This theorem is referenced by: (None)
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