Theorem domepOLD 5794

Description: Obsolete proof of dmep 5793 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 2019	. . . 4 ⊢ 𝑥 = 𝑥
2		el 5270	. . . . 5 ⊢ ∃𝑦 𝑥 ∈ 𝑦
3		epel 5469	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4	3	exbii 1848	. . . . 5 ⊢ (∃𝑦 𝑥 E 𝑦 ↔ ∃𝑦 𝑥 ∈ 𝑦)
5	2, 4	mpbir 233	. . . 4 ⊢ ∃𝑦 𝑥 E 𝑦
6	1, 5	2th 266	. . 3 ⊢ (𝑥 = 𝑥 ↔ ∃𝑦 𝑥 E 𝑦)
7	6	abbii 2886	. 2 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
8		df-v 3496	. 2 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
9		df-dm 5565	. 2 ⊢ dom E = {𝑥 ∣ ∃𝑦 𝑥 E 𝑦}
10	7, 8, 9	3eqtr4ri 2855	1 ⊢ dom E = V

Syntax hints:   = wceq 1537  ∃wex 1780  {cab 2799  Vcvv 3494   class class class wbr 5066   E cep 5464  dom cdm 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-eprel 5465  df-dm 5565

This theorem is referenced by: (None)