Theorem fvray 32248

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 6653	. 2 ⊢ (𝑃Ray𝐴) = (Ray‘⟨𝑃, 𝐴⟩)
2		eqid 2622	. . . . 5 ⊢ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}
3		fveq2 6191	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
4	3	eleq2d 2687	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝑃 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑁)))
5	3	eleq2d 2687	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
6	4, 5	3anbi12d 1400	. . . . . . 7 ⊢ (𝑛 = 𝑁 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)))
7		rabeq 3192	. . . . . . . . 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 2632	. . . . . . 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 747	. . . . . 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 3309	. . . . 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 720	. . . 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 1067	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → 𝑃 ∈ (𝔼‘𝑁))
14		simpr2 1068	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → 𝐴 ∈ (𝔼‘𝑁))
15		fvex 6201	. . . . . . 7 ⊢ (𝔼‘𝑁) ∈ V
16	15	rabex 4813	. . . . . 6 ⊢ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ∈ V
17		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑛)))
18		neeq1 2856	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝 ≠ 𝑎 ↔ 𝑃 ≠ 𝑎))
19	17, 18	3anbi13d 1401	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝 ≠ 𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎)))
20		breq1 4656	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑃 → (𝑝OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝑎, 𝑥⟩))
21	20	rabbidv 3189	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})
22	21	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}))
23	19, 22	anbi12d 747	. . . . . . . 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 3052	. . . . . . 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 2689	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
26		neeq2 2857	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑃 ≠ 𝑎 ↔ 𝑃 ≠ 𝐴))
27	25, 26	3anbi23d 1402	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃 ≠ 𝐴)))
28		opeq1 4402	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
29	28	breq2d 4665	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑃OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝐴, 𝑥⟩))
30	29	rabbidv 3189	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
31	30	eqeq2d 2632	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
32	27, 31	anbi12d 747	. . . . . . . 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 3052	. . . . . . 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 2626	. . . . . . . . 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 740	. . . . . . . 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 3052	. . . . . . 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 6748	. . . . . 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 1413	. . . . 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 693	. . . 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 247	. . 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 4654	. . . . 5 ⊢ (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ↔ ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ Ray)
42		df-ray 32245	. . . . . 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 2693	. . . . 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 264	. . . 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 32247	. . . . 5 ⊢ Fun Ray
46		funbrfv 6234	. . . . 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 225	. . 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 2668	1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃 ≠ 𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  {crab 2916  Vcvv 3200  ⟨cop 4183   class class class wbr 4653  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650  {coprab 6651  ℕcn 11020  𝔼cee 25768  OutsideOfcoutsideof 32226  Raycray 32242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-z 11378  df-uz 11688  df-fz 12327  df-ee 25771  df-ray 32245

This theorem is referenced by:  lineunray  32254