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Theorem funray 35635
Description: Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funray Fun Ray

Proof of Theorem funray
Dummy variables π‘š π‘Ž 𝑛 𝑝 π‘Ÿ 𝑠 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reeanv 3218 . . . . . 6 (βˆƒπ‘› ∈ β„• βˆƒπ‘š ∈ 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‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 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‘₯⟩}) ∧ βˆƒπ‘š ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})))
2 simp1 1133 . . . . . . . . . . 11 ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) β†’ 𝑝 ∈ (π”Όβ€˜π‘›))
3 simp1 1133 . . . . . . . . . . 11 ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) β†’ 𝑝 ∈ (π”Όβ€˜π‘š))
4 axdimuniq 28665 . . . . . . . . . . . . . . 15 (((𝑛 ∈ β„• ∧ 𝑝 ∈ (π”Όβ€˜π‘›)) ∧ (π‘š ∈ β„• ∧ 𝑝 ∈ (π”Όβ€˜π‘š))) β†’ 𝑛 = π‘š)
5 fveq2 6882 . . . . . . . . . . . . . . . . . . 19 (𝑛 = π‘š β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘š))
6 rabeq 3438 . . . . . . . . . . . . . . . . . . 19 ((π”Όβ€˜π‘›) = (π”Όβ€˜π‘š) β†’ {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})
75, 6syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = π‘š β†’ {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})
87eqeq2d 2735 . . . . . . . . . . . . . . . . 17 (𝑛 = π‘š β†’ (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ↔ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}))
98anbi1d 629 . . . . . . . . . . . . . . . 16 (𝑛 = π‘š β†’ ((π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) ↔ (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})))
10 eqtr3 2750 . . . . . . . . . . . . . . . 16 ((π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) β†’ π‘Ÿ = 𝑠)
119, 10syl6bi 253 . . . . . . . . . . . . . . 15 (𝑛 = π‘š β†’ ((π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) β†’ π‘Ÿ = 𝑠))
124, 11syl 17 . . . . . . . . . . . . . 14 (((𝑛 ∈ β„• ∧ 𝑝 ∈ (π”Όβ€˜π‘›)) ∧ (π‘š ∈ β„• ∧ 𝑝 ∈ (π”Όβ€˜π‘š))) β†’ ((π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) β†’ π‘Ÿ = 𝑠))
1312an4s 657 . . . . . . . . . . . . 13 (((𝑛 ∈ β„• ∧ π‘š ∈ β„•) ∧ (𝑝 ∈ (π”Όβ€˜π‘›) ∧ 𝑝 ∈ (π”Όβ€˜π‘š))) β†’ ((π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) β†’ π‘Ÿ = 𝑠))
1413ex 412 . . . . . . . . . . . 12 ((𝑛 ∈ β„• ∧ π‘š ∈ β„•) β†’ ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ 𝑝 ∈ (π”Όβ€˜π‘š)) β†’ ((π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) β†’ π‘Ÿ = 𝑠)))
1514com3l 89 . . . . . . . . . . 11 ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ 𝑝 ∈ (π”Όβ€˜π‘š)) β†’ ((π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) β†’ ((𝑛 ∈ β„• ∧ π‘š ∈ β„•) β†’ π‘Ÿ = 𝑠)))
162, 3, 15syl2an 595 . . . . . . . . . 10 (((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ (𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž)) β†’ ((π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) β†’ ((𝑛 ∈ β„• ∧ π‘š ∈ β„•) β†’ π‘Ÿ = 𝑠)))
1716imp 406 . . . . . . . . 9 ((((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ (𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž)) ∧ (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})) β†’ ((𝑛 ∈ β„• ∧ π‘š ∈ β„•) β†’ π‘Ÿ = 𝑠))
1817an4s 657 . . . . . . . 8 ((((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‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})) β†’ ((𝑛 ∈ β„• ∧ π‘š ∈ β„•) β†’ π‘Ÿ = 𝑠))
1918com12 32 . . . . . . 7 ((𝑛 ∈ β„• ∧ π‘š ∈ β„•) 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‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})) β†’ π‘Ÿ = 𝑠))
2019rexlimivv 3191 . . . . . 6 (βˆƒπ‘› ∈ β„• βˆƒπ‘š ∈ 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‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})) β†’ π‘Ÿ = 𝑠)
211, 20sylbir 234 . . . . 5 ((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‘š ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})) β†’ π‘Ÿ = 𝑠)
2221gen2 1790 . . . 4 βˆ€π‘Ÿβˆ€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‘₯⟩}) ∧ βˆƒπ‘š ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})) β†’ π‘Ÿ = 𝑠)
23 eqeq1 2728 . . . . . . . 8 (π‘Ÿ = 𝑠 β†’ (π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ↔ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}))
2423anbi2d 628 . . . . . . 7 (π‘Ÿ = 𝑠 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) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})))
2524rexbidv 3170 . . . . . 6 (π‘Ÿ = 𝑠 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) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})))
265eleq2d 2811 . . . . . . . . 9 (𝑛 = π‘š β†’ (𝑝 ∈ (π”Όβ€˜π‘›) ↔ 𝑝 ∈ (π”Όβ€˜π‘š)))
275eleq2d 2811 . . . . . . . . 9 (𝑛 = π‘š β†’ (π‘Ž ∈ (π”Όβ€˜π‘›) ↔ π‘Ž ∈ (π”Όβ€˜π‘š)))
2826, 273anbi12d 1433 . . . . . . . 8 (𝑛 = π‘š β†’ ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ↔ (𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž)))
297eqeq2d 2735 . . . . . . . 8 (𝑛 = π‘š β†’ (𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩} ↔ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}))
3028, 29anbi12d 630 . . . . . . 7 (𝑛 = π‘š β†’ (((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) ↔ ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})))
3130cbvrexvw 3227 . . . . . 6 (βˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) ↔ βˆƒπ‘š ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}))
3225, 31bitrdi 287 . . . . 5 (π‘Ÿ = 𝑠 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‘š ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})))
3332mo4 2552 . . . 4 (βˆƒ*π‘Ÿβˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩}) ↔ βˆ€π‘Ÿβˆ€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‘₯⟩}) ∧ βˆƒπ‘š ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) ∧ 𝑠 = {π‘₯ ∈ (π”Όβ€˜π‘š) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})) β†’ π‘Ÿ = 𝑠))
3422, 33mpbir 230 . . 3 βˆƒ*π‘Ÿβˆƒπ‘› ∈ β„• ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) ∧ π‘Ÿ = {π‘₯ ∈ (π”Όβ€˜π‘›) ∣ 𝑝OutsideOfβŸ¨π‘Ž, π‘₯⟩})
3534funoprab 7523 . 2 Fun {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‘₯⟩})}
36 df-ray 35633 . . 3 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‘₯⟩})}
3736funeqi 6560 . 2 (Fun Ray ↔ Fun {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‘₯⟩})})
3835, 37mpbir 230 1 Fun Ray
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084  βˆ€wal 1531   = wceq 1533   ∈ wcel 2098  βˆƒ*wmo 2524   β‰  wne 2932  βˆƒwrex 3062  {crab 3424  βŸ¨cop 4627   class class class wbr 5139  Fun wfun 6528  β€˜cfv 6534  {coprab 7403  β„•cn 12211  π”Όcee 28640  OutsideOfcoutsideof 35614  Raycray 35630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-z 12558  df-uz 12822  df-fz 13486  df-ee 28643  df-ray 35633
This theorem is referenced by:  fvray  35636
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