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Theorem funray 35736
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 3223 . . . . . 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 1134 . . . . . . . . . . 11 ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) β†’ 𝑝 ∈ (π”Όβ€˜π‘›))
3 simp1 1134 . . . . . . . . . . 11 ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) β†’ 𝑝 ∈ (π”Όβ€˜π‘š))
4 axdimuniq 28737 . . . . . . . . . . . . . . 15 (((𝑛 ∈ β„• ∧ 𝑝 ∈ (π”Όβ€˜π‘›)) ∧ (π‘š ∈ β„• ∧ 𝑝 ∈ (π”Όβ€˜π‘š))) β†’ 𝑛 = π‘š)
5 fveq2 6897 . . . . . . . . . . . . . . . . . . 19 (𝑛 = π‘š β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘š))
6 rabeq 3443 . . . . . . . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . 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 2754 . . . . . . . . . . . . . . . 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, 10biimtrdi 252 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . 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 659 . . . . . . . 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 3196 . . . . . 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 1791 . . . 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 2732 . . . . . . . 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 629 . . . . . . 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 3175 . . . . . 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 2815 . . . . . . . . 9 (𝑛 = π‘š β†’ (𝑝 ∈ (π”Όβ€˜π‘›) ↔ 𝑝 ∈ (π”Όβ€˜π‘š)))
275eleq2d 2815 . . . . . . . . 9 (𝑛 = π‘š β†’ (π‘Ž ∈ (π”Όβ€˜π‘›) ↔ π‘Ž ∈ (π”Όβ€˜π‘š)))
2826, 273anbi12d 1434 . . . . . . . 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 2739 . . . . . . . 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 631 . . . . . . 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 3232 . . . . . 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 2556 . . . 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 7542 . 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 35734 . . 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 6574 . 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 1085  βˆ€wal 1532   = wceq 1534   ∈ wcel 2099  βˆƒ*wmo 2528   β‰  wne 2937  βˆƒwrex 3067  {crab 3429  βŸ¨cop 4635   class class class wbr 5148  Fun wfun 6542  β€˜cfv 6548  {coprab 7421  β„•cn 12243  π”Όcee 28712  OutsideOfcoutsideof 35715  Raycray 35731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-nn 12244  df-z 12590  df-uz 12854  df-fz 13518  df-ee 28715  df-ray 35734
This theorem is referenced by:  fvray  35737
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