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Theorem funray 34778
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 3216 . . . . . 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 1137 . . . . . . . . . . 11 ((𝑝 ∈ (π”Όβ€˜π‘›) ∧ π‘Ž ∈ (π”Όβ€˜π‘›) ∧ 𝑝 β‰  π‘Ž) β†’ 𝑝 ∈ (π”Όβ€˜π‘›))
3 simp1 1137 . . . . . . . . . . 11 ((𝑝 ∈ (π”Όβ€˜π‘š) ∧ π‘Ž ∈ (π”Όβ€˜π‘š) ∧ 𝑝 β‰  π‘Ž) β†’ 𝑝 ∈ (π”Όβ€˜π‘š))
4 axdimuniq 27911 . . . . . . . . . . . . . . 15 (((𝑛 ∈ β„• ∧ 𝑝 ∈ (π”Όβ€˜π‘›)) ∧ (π‘š ∈ β„• ∧ 𝑝 ∈ (π”Όβ€˜π‘š))) β†’ 𝑛 = π‘š)
5 fveq2 6846 . . . . . . . . . . . . . . . . . . 19 (𝑛 = π‘š β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘š))
6 rabeq 3420 . . . . . . . . . . . . . . . . . . 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 2744 . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . 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 2759 . . . . . . . . . . . . . . . 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 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 414 . . . . . . . . . . . 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 597 . . . . . . . . . 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 408 . . . . . . . . 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 3193 . . . . . 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 1799 . . . 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 2737 . . . . . . . 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 630 . . . . . . 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 3172 . . . . . 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 2820 . . . . . . . . 9 (𝑛 = π‘š β†’ (𝑝 ∈ (π”Όβ€˜π‘›) ↔ 𝑝 ∈ (π”Όβ€˜π‘š)))
275eleq2d 2820 . . . . . . . . 9 (𝑛 = π‘š β†’ (π‘Ž ∈ (π”Όβ€˜π‘›) ↔ π‘Ž ∈ (π”Όβ€˜π‘š)))
2826, 273anbi12d 1438 . . . . . . . 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 2744 . . . . . . . 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 632 . . . . . . 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 3225 . . . . . 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 2561 . . . 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 7482 . 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 34776 . . 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 6526 . 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 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  βˆƒ*wmo 2533   β‰  wne 2940  βˆƒwrex 3070  {crab 3406  βŸ¨cop 4596   class class class wbr 5109  Fun wfun 6494  β€˜cfv 6500  {coprab 7362  β„•cn 12161  π”Όcee 27886  OutsideOfcoutsideof 34757  Raycray 34773
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-z 12508  df-uz 12772  df-fz 13434  df-ee 27889  df-ray 34776
This theorem is referenced by:  fvray  34779
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