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Theorem fvray 31890
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) ∧ 𝑃𝐴)) → (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.
StepHypRef Expression
1 df-ov 6607 . 2 (𝑃Ray𝐴) = (Ray‘⟨𝑃, 𝐴⟩)
2 eqid 2621 . . . . 5 {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}
3 fveq2 6148 . . . . . . . . 9 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
43eleq2d 2684 . . . . . . . 8 (𝑛 = 𝑁 → (𝑃 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑁)))
53eleq2d 2684 . . . . . . . 8 (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
64, 53anbi12d 1397 . . . . . . 7 (𝑛 = 𝑁 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ↔ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)))
7 rabeq 3179 . . . . . . . . 9 ((𝔼‘𝑛) = (𝔼‘𝑁) → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
83, 7syl 17 . . . . . . . 8 (𝑛 = 𝑁 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
98eqeq2d 2631 . . . . . . 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⟩}))
106, 9anbi12d 746 . . . . . 6 (𝑛 = 𝑁 → (((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}) ↔ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
1110rspcev 3295 . . . . 5 ((𝑁 ∈ ℕ ∧ ((𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})) → ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
122, 11mpanr2 719 . . . 4 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)) → ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
13 simpr1 1065 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)) → 𝑃 ∈ (𝔼‘𝑁))
14 simpr2 1066 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)) → 𝐴 ∈ (𝔼‘𝑁))
15 fvex 6158 . . . . . . 7 (𝔼‘𝑁) ∈ V
1615rabex 4773 . . . . . 6 {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ∈ V
17 eleq1 2686 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝 ∈ (𝔼‘𝑛) ↔ 𝑃 ∈ (𝔼‘𝑛)))
18 neeq1 2852 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑎𝑃𝑎))
1917, 183anbi13d 1398 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃𝑎)))
20 breq1 4616 . . . . . . . . . . 11 (𝑝 = 𝑃 → (𝑝OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝑎, 𝑥⟩))
2120rabbidv 3177 . . . . . . . . . 10 (𝑝 = 𝑃 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})
2221eqeq2d 2631 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}))
2319, 22anbi12d 746 . . . . . . . 8 (𝑝 = 𝑃 → (((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})))
2423rexbidv 3045 . . . . . . 7 (𝑝 = 𝑃 → (∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩})))
25 eleq1 2686 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
26 neeq2 2853 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑃𝑎𝑃𝐴))
2725, 263anbi23d 1399 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃𝑎) ↔ (𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴)))
28 opeq1 4370 . . . . . . . . . . . 12 (𝑎 = 𝐴 → ⟨𝑎, 𝑥⟩ = ⟨𝐴, 𝑥⟩)
2928breq2d 4625 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑃OutsideOf⟨𝑎, 𝑥⟩ ↔ 𝑃OutsideOf⟨𝐴, 𝑥⟩))
3029rabbidv 3177 . . . . . . . . . 10 (𝑎 = 𝐴 → {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
3130eqeq2d 2631 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩} ↔ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
3227, 31anbi12d 746 . . . . . . . 8 (𝑎 = 𝐴 → (((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}) ↔ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
3332rexbidv 3045 . . . . . . 7 (𝑎 = 𝐴 → (∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑃𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝑎, 𝑥⟩}) ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
34 eqeq1 2625 . . . . . . . . 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⟩}))
3534anbi2d 739 . . . . . . . 8 (𝑟 = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}) ↔ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
3635rexbidv 3045 . . . . . . 7 (𝑟 = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}) ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
3724, 33, 36eloprabg 6701 . . . . . 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) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})} ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
3816, 37mp3an3 1410 . . . . 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) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})} ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
3913, 14, 38syl2anc 692 . . . 4 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)) → (⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})} ↔ ∃𝑛 ∈ ℕ ((𝑃 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛) ∧ 𝑃𝐴) ∧ {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})))
4012, 39mpbird 247 . . 3 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)) → ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})})
41 df-br 4614 . . . . 5 (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} ↔ ⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ Ray)
42 df-ray 31887 . . . . . 6 Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
4342eleq2i 2690 . . . . 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) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})})
4441, 43bitri 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) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})})
45 funray 31889 . . . . 5 Fun Ray
46 funbrfv 6191 . . . . 5 (Fun Ray → (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (Ray‘⟨𝑃, 𝐴⟩) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}))
4745, 46ax-mp 5 . . . 4 (⟨𝑃, 𝐴⟩Ray{𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩} → (Ray‘⟨𝑃, 𝐴⟩) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
4844, 47sylbir 225 . . 3 (⟨⟨𝑃, 𝐴⟩, {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩}⟩ ∈ {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})} → (Ray‘⟨𝑃, 𝐴⟩) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
4940, 48syl 17 . 2 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)) → (Ray‘⟨𝑃, 𝐴⟩) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
501, 49syl5eq 2667 1 ((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  {crab 2911  Vcvv 3186  cop 4154   class class class wbr 4613  Fun wfun 5841  cfv 5847  (class class class)co 6604  {coprab 6605  cn 10964  𝔼cee 25668  OutsideOfcoutsideof 31868  Raycray 31884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-z 11322  df-uz 11632  df-fz 12269  df-ee 25671  df-ray 31887
This theorem is referenced by:  lineunray  31896
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