Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvray Structured version   Visualization version   GIF version

Theorem fvray 35113
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 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)) β†’ (𝑃Ray𝐴) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝑁   π‘₯,𝑃

Proof of Theorem fvray
Dummy variables π‘Ž 𝑛 𝑝 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7412 . 2 (𝑃Ray𝐴) = (Rayβ€˜βŸ¨π‘ƒ, 𝐴⟩)
2 eqid 2733 . . . . 5 {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}
3 fveq2 6892 . . . . . . . . 9 (𝑛 = 𝑁 β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
43eleq2d 2820 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝑃 ∈ (π”Όβ€˜π‘›) ↔ 𝑃 ∈ (π”Όβ€˜π‘)))
53eleq2d 2820 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝐴 ∈ (π”Όβ€˜π‘›) ↔ 𝐴 ∈ (π”Όβ€˜π‘)))
64, 53anbi12d 1438 . . . . . . 7 (𝑛 = 𝑁 β†’ ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ↔ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)))
7 rabeq 3447 . . . . . . . . 9 ((π”Όβ€˜π‘›) = (π”Όβ€˜π‘) β†’ {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})
83, 7syl 17 . . . . . . . 8 (𝑛 = 𝑁 β†’ {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})
98eqeq2d 2744 . . . . . . 7 (𝑛 = 𝑁 β†’ ({π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} ↔ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}))
106, 9anbi12d 632 . . . . . 6 (𝑛 = 𝑁 β†’ (((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}) ↔ ((𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})))
1110rspcev 3613 . . . . 5 ((𝑁 ∈ β„• ∧ ((𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})) β†’ βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}))
122, 11mpanr2 703 . . . 4 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)) β†’ βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}))
13 simpr1 1195 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
14 simpr2 1196 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
15 fvex 6905 . . . . . . 7 (π”Όβ€˜π‘) ∈ V
1615rabex 5333 . . . . . 6 {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} ∈ V
17 eleq1 2822 . . . . . . . . . 10 (𝑝 = 𝑃 β†’ (𝑝 ∈ (π”Όβ€˜π‘›) ↔ 𝑃 ∈ (π”Όβ€˜π‘›)))
18 neeq1 3004 . . . . . . . . . 10 (𝑝 = 𝑃 β†’ (𝑝 β‰  π‘Ž ↔ 𝑃 β‰  π‘Ž))
1917, 183anbi13d 1439 . . . . . . . . 9 (𝑝 = 𝑃 β†’ ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ↔ (𝑃 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  π‘Ž)))
20 breq1 5152 . . . . . . . . . . 11 (𝑝 = 𝑃 β†’ (𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩ ↔ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩))
2120rabbidv 3441 . . . . . . . . . 10 (𝑝 = 𝑃 β†’ {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩})
2221eqeq2d 2744 . . . . . . . . 9 (𝑝 = 𝑃 β†’ (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ↔ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩}))
2319, 22anbi12d 632 . . . . . . . 8 (𝑝 = 𝑃 β†’ (((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) ↔ ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩})))
2423rexbidv 3179 . . . . . . 7 (𝑝 = 𝑃 β†’ (βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) ↔ βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩})))
25 eleq1 2822 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ (π‘Ž ∈ (π”Όβ€˜π‘›) ↔ 𝐴 ∈ (π”Όβ€˜π‘›)))
26 neeq2 3005 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ (𝑃 β‰  π‘Ž ↔ 𝑃 β‰  𝐴))
2725, 263anbi23d 1440 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  π‘Ž) ↔ (𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴)))
28 opeq1 4874 . . . . . . . . . . . 12 (π‘Ž = 𝐴 β†’ βŸ¨π‘Ž, π‘₯⟩ = ⟨𝐴, π‘₯⟩)
2928breq2d 5161 . . . . . . . . . . 11 (π‘Ž = 𝐴 β†’ (𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩ ↔ 𝑃OutsideOf⟨𝐴, π‘₯⟩))
3029rabbidv 3441 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})
3130eqeq2d 2744 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩} ↔ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}))
3227, 31anbi12d 632 . . . . . . . 8 (π‘Ž = 𝐴 β†’ (((𝑃 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩}) ↔ ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})))
3332rexbidv 3179 . . . . . . 7 (π‘Ž = 𝐴 β†’ (βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOfβŸ¨π‘Ž, π‘₯⟩}) ↔ βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})))
34 eqeq1 2737 . . . . . . . . 9 (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} β†’ (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} ↔ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}))
3534anbi2d 630 . . . . . . . 8 (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} β†’ (((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}) ↔ ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})))
3635rexbidv 3179 . . . . . . 7 (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} β†’ (βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}) ↔ βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})))
3724, 33, 36eloprabg 7518 . . . . . 6 ((𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} ∈ V) β†’ (βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ {βŸ¨βŸ¨π‘, π‘ŽβŸ©, π‘ŸβŸ© ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})} ↔ βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})))
3816, 37mp3an3 1451 . . . . 5 ((𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)) β†’ (βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ {βŸ¨βŸ¨π‘, π‘ŽβŸ©, π‘ŸβŸ© ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})} ↔ βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})))
3913, 14, 38syl2anc 585 . . . 4 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)) β†’ (βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ {βŸ¨βŸ¨π‘, π‘ŽβŸ©, π‘ŸβŸ© ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})} ↔ βˆƒπ‘› ∈ β„• ((𝑃 ∈ (π”Όβ€˜π‘›) ∧ 𝐴 ∈ (π”Όβ€˜π‘›) ∧ 𝑃 β‰  𝐴) ∧ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})))
4012, 39mpbird 257 . . 3 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)) β†’ βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ {βŸ¨βŸ¨π‘, π‘ŽβŸ©, π‘ŸβŸ© ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})})
41 df-br 5150 . . . . 5 (βŸ¨π‘ƒ, 𝐴⟩Ray{π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} ↔ βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ Ray)
42 df-ray 35110 . . . . . 6 Ray = {βŸ¨βŸ¨π‘, π‘ŽβŸ©, π‘ŸβŸ© ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})}
4342eleq2i 2826 . . . . 5 (βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ Ray ↔ βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ {βŸ¨βŸ¨π‘, π‘ŽβŸ©, π‘ŸβŸ© ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})})
4441, 43bitri 275 . . . 4 (βŸ¨π‘ƒ, 𝐴⟩Ray{π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} ↔ βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ {βŸ¨βŸ¨π‘, π‘ŽβŸ©, π‘ŸβŸ© ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})})
45 funray 35112 . . . . 5 Fun Ray
46 funbrfv 6943 . . . . 5 (Fun Ray β†’ (βŸ¨π‘ƒ, 𝐴⟩Ray{π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} β†’ (Rayβ€˜βŸ¨π‘ƒ, 𝐴⟩) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}))
4745, 46ax-mp 5 . . . 4 (βŸ¨π‘ƒ, 𝐴⟩Ray{π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩} β†’ (Rayβ€˜βŸ¨π‘ƒ, 𝐴⟩) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})
4844, 47sylbir 234 . . 3 (βŸ¨βŸ¨π‘ƒ, 𝐴⟩, {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩}⟩ ∈ {βŸ¨βŸ¨π‘, π‘ŽβŸ©, π‘ŸβŸ© ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})} β†’ (Rayβ€˜βŸ¨π‘ƒ, 𝐴⟩) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})
4940, 48syl 17 . 2 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)) β†’ (Rayβ€˜βŸ¨π‘ƒ, 𝐴⟩) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})
501, 49eqtrid 2785 1 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝐴)) β†’ (𝑃Ray𝐴) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOf⟨𝐴, π‘₯⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071  {crab 3433  Vcvv 3475  βŸ¨cop 4635   class class class wbr 5149  Fun wfun 6538  β€˜cfv 6544  (class class class)co 7409  {coprab 7410  β„•cn 12212  π”Όcee 28146  OutsideOfcoutsideof 35091  Raycray 35107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-z 12559  df-uz 12823  df-fz 13485  df-ee 28149  df-ray 35110
This theorem is referenced by:  lineunray  35119
  Copyright terms: Public domain W3C validator