Theorem fvray 32564

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 6873	. 2 ⊢ (𝑃Ray𝐴) = (Ray‘⟨𝑃, 𝐴⟩)
2		eqid 2806	. . . . 5 ⊢ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}
3		fveq2 6404	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
4	3	eleq2d 2871	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑃 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑁)))
5	3	eleq2d 2871	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
6	4, 5	3anbi12d 1554	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)))
7		rabeq 3382	. . . . . . . . 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 2816	. . . . . . 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 618	. . . . . 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 3502	. . . . 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 687	. . . 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 1241	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → 𝑃 ∈ (𝔼‘𝑁))
14		simpr2 1243	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → 𝐴 ∈ (𝔼‘𝑁))
15		fvex 6417	. . . . . . 7 ⊢ (𝔼‘𝑁) ∈ V
16	15	rabex 5007	. . . . . 6 ⊢ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ∈ V
17		eleq1 2873	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑛)))
18		neeq1 3040	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝 ≠ 𝑎 ↔ 𝑃 ≠ 𝑎))
19	17, 18	3anbi13d 1555	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎)))
20		breq1 4847	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → (𝑝OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝑎, 𝑥⟩))
21	20	rabbidv 3379	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})
22	21	eqeq2d 2816	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}))
23	19, 22	anbi12d 618	. . . . . . . 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 3240	. . . . . . 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 2873	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
26		neeq2 3041	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑃 ≠ 𝑎 ↔ 𝑃 ≠ 𝐴))
27	25, 26	3anbi23d 1556	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴)))
28		opeq1 4595	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
29	28	breq2d 4856	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑃OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝐴, 𝑥⟩))
30	29	rabbidv 3379	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
31	30	eqeq2d 2816	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
32	27, 31	anbi12d 618	. . . . . . . 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 3240	. . . . . . 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 2810	. . . . . . . . 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 616	. . . . . . . 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 3240	. . . . . . 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 6974	. . . . . 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 1567	. . . . 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 575	. . . 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 248	. . 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 4845	. . . . 5 ⊢ (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ↔ ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ Ray)
42		df-ray 32561	. . . . . 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 2877	. . . . 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 266	. . . 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 32563	. . . . 5 ⊢ Fun Ray
46		funbrfv 6450	. . . . 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 226	. . 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	syl5eq 2852	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1100   = wceq 1637   ∈ wcel 2156   ≠ wne 2978  ∃wrex 3097  {crab 3100  Vcvv 3391  ⟨cop 4376   class class class wbr 4844  Fun wfun 6091  ‘cfv 6097  (class class class)co 6870  {coprab 6871  ℕcn 11301  𝔼cee 25978  OutsideOfcoutsideof 32542  Raycray 32558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-er 7975  df-map 8090  df-en 8189  df-dom 8190  df-sdom 8191  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-nn 11302  df-z 11640  df-uz 11901  df-fz 12546  df-ee 25981  df-ray 32561

This theorem is referenced by:  lineunray  32570