Theorem fvray 36455

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 7395	. 2 ⊢ (𝑃Ray𝐴) = (Ray‘⟨𝑃, 𝐴⟩)
2		eqid 2761	. . . . 5 ⊢ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}
3		fveq2 6863	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
4	3	eleq2d 2847	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑃 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑁)))
5	3	eleq2d 2847	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
6	4, 5	3anbi12d 1457	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)))
7		rabeq 3427	. . . . . . . . 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 2772	. . . . . . 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 641	. . . . . 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 3581	. . . . 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 714	. . . 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 1207	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → 𝑃 ∈ (𝔼‘𝑁))
14		simpr2 1208	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → 𝐴 ∈ (𝔼‘𝑁))
15		fvex 6876	. . . . . . 7 ⊢ (𝔼‘𝑁) ∈ V
16	15	rabex 5294	. . . . . 6 ⊢ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ∈ V
17		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑛)))
18		neeq1 3018	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝 ≠ 𝑎 ↔ 𝑃 ≠ 𝑎))
19	17, 18	3anbi13d 1458	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎)))
20		breq1 5102	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → (𝑝OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝑎, 𝑥⟩))
21	20	rabbidv 3420	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})
22	21	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}))
23	19, 22	anbi12d 641	. . . . . . . 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 3185	. . . . . . 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 2849	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
26		neeq2 3019	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑃 ≠ 𝑎 ↔ 𝑃 ≠ 𝐴))
27	25, 26	3anbi23d 1459	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴)))
28		opeq1 4830	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
29	28	breq2d 5111	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑃OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝐴, 𝑥⟩))
30	29	rabbidv 3420	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
31	30	eqeq2d 2772	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
32	27, 31	anbi12d 641	. . . . . . . 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 3185	. . . . . . 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 2765	. . . . . . . . 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 639	. . . . . . . 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 3185	. . . . . . 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 7502	. . . . . 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 1470	. . . . 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 593	. . . 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 259	. . 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 5100	. . . . 5 ⊢ (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ↔ ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ Ray)
42		df-ray 36452	. . . . . 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 2853	. . . . 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 277	. . . 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 36454	. . . . 5 ⊢ Fun Ray
46		funbrfv 6911	. . . . 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 237	. . 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 2808	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∃wrex 3085  {crab 3413  Vcvv 3453  ⟨cop 4587   class class class wbr 5099  Fun wfun 6511  ‘cfv 6517  (class class class)co 7392  {coprab 7393  ℕcn 12207  𝔼cee 29034  OutsideOfcoutsideof 36433  Raycray 36449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-z 12566  df-uz 12837  df-fz 13510  df-ee 29037  df-ray 36452

This theorem is referenced by:  lineunray  36461