Theorem fvray 36354

Description: Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fvray	⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})

Distinct variable groups:   𝑥,𝐴   𝑥,𝑁   𝑥,𝑃

Proof of Theorem fvray

Dummy variables 𝑎 𝑛 𝑝 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7371	. 2 ⊢ (𝑃Ray𝐴) = (Ray‘⟨𝑃, 𝐴⟩)
2		eqid 2737	. . . . 5 ⊢ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}
3		fveq2 6842	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
4	3	eleq2d 2823	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑃 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑁)))
5	3	eleq2d 2823	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
6	4, 5	3anbi12d 1440	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)))
7		rabeq 3415	. . . . . . . . 9 ⊢ ((𝔼‘𝑛) = (𝔼‘𝑁) → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
8	3, 7	syl 17	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
9	8	eqeq2d 2748	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ({𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ↔ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
10	6, 9	anbi12d 633	. . . . . 6 ⊢ (𝑛 = 𝑁 → (((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}) ↔ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
11	10	rspcev 3578	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})) → ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
12	2, 11	mpanr2 705	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
13		simpr1 1196	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → 𝑃 ∈ (𝔼‘𝑁))
14		simpr2 1197	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → 𝐴 ∈ (𝔼‘𝑁))
15		fvex 6855	. . . . . . 7 ⊢ (𝔼‘𝑁) ∈ V
16	15	rabex 5286	. . . . . 6 ⊢ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ∈ V
17		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑛)))
18		neeq1 2995	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝 ≠ 𝑎 ↔ 𝑃 ≠ 𝑎))
19	17, 18	3anbi13d 1441	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎)))
20		breq1 5103	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → (𝑝OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝑎, 𝑥⟩))
21	20	rabbidv 3408	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})
22	21	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}))
23	19, 22	anbi12d 633	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})))
24	23	rexbidv 3162	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})))
25		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
26		neeq2 2996	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑃 ≠ 𝑎 ↔ 𝑃 ≠ 𝐴))
27	25, 26	3anbi23d 1442	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴)))
28		opeq1 4831	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
29	28	breq2d 5112	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑃OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝐴, 𝑥⟩))
30	29	rabbidv 3408	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
31	30	eqeq2d 2748	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
32	27, 31	anbi12d 633	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}) ↔ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
33	32	rexbidv 3162	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
34		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑟 = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ↔ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
35	34	anbi2d 631	. . . . . . . 8 ⊢ (𝑟 = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}) ↔ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
36	35	rexbidv 3162	. . . . . . 7 ⊢ (𝑟 = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}) ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
37	24, 33, 36	eloprabg 7478	. . . . . 6 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ∈ V) → (⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})} ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
38	16, 37	mp3an3 1453	. . . . 5 ⊢ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})} ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
39	13, 14, 38	syl2anc 585	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})} ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
40	12, 39	mpbird 257	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})})
41		df-br 5101	. . . . 5 ⊢ (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ↔ ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ Ray)
42		df-ray 36351	. . . . . 6 ⊢ Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
43	42	eleq2i 2829	. . . . 5 ⊢ (⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ Ray ↔ ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})})
44	41, 43	bitri 275	. . . 4 ⊢ (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ↔ ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})})
45		funray 36353	. . . . 5 ⊢ Fun Ray
46		funbrfv 6890	. . . . 5 ⊢ (Fun Ray → (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (Ray‘⟨𝑃, 𝐴⟩) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
47	45, 46	ax-mp 5	. . . 4 ⊢ (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (Ray‘⟨𝑃, 𝐴⟩) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
48	44, 47	sylbir 235	. . 3 ⊢ (⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})} → (Ray‘⟨𝑃, 𝐴⟩) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
49	40, 48	syl 17	. 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (Ray‘⟨𝑃, 𝐴⟩) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
50	1, 49	eqtrid 2784	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  {crab 3401  Vcvv 3442  ⟨cop 4588   class class class wbr 5100  Fun wfun 6494  ‘cfv 6500  (class class class)co 7368  {coprab 7369  ℕcn 12157  𝔼cee 28972  OutsideOfcoutsideof 36332  Raycray 36348

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-z 12501  df-uz 12764  df-fz 13436  df-ee 28975  df-ray 36351

This theorem is referenced by:  lineunray  36360