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Theorem domepOLD 5793
Description: Obsolete proof of dmep 5792 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 2020 . . . 4 𝑥 = 𝑥
2 el 5262 . . . . 5 𝑦 𝑥𝑦
3 epel 5463 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
43exbii 1855 . . . . 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 2808 . 2 {𝑥𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8 df-v 3410 . 2 V = {𝑥𝑥 = 𝑥}
9 df-dm 5561 . 2 dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
107, 8, 93eqtr4ri 2776 1 dom E = V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wex 1787  {cab 2714  Vcvv 3408   class class class wbr 5053   E cep 5459  dom cdm 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-eprel 5460  df-dm 5561
This theorem is referenced by: (None)
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