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.

Step	Hyp	Ref	Expression
1		equid 2020	. . . 4 ⊢ 𝑥 = 𝑥
2		el 5262	. . . . 5 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5463	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1855	. . . . 5 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 234	. . . 4 ⊢ ∃𝑦 𝑥 E 𝑦
6	1, 5	2th 267	. . 3 ⊢ (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦)
7	6	abbii 2808	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8		df-v 3410	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9		df-dm 5561	. 2 ⊢ dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
10	7, 8, 9	3eqtr4ri 2776	1 ⊢ dom E = V

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)