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Theorem pthdivtx 28774
Description: The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)
Assertion
Ref Expression
pthdivtx ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ (𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (0...(β™―β€˜πΉ)) ∧ 𝐼 β‰  𝐽)) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))
Proof of Theorem pthdivtx
StepHypRef Expression
1 ispth 28768 . . 3 (𝐹(Pathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…))
2 trliswlk 28742 . . . . 5 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 eqid 2731 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
43wlkp 28661 . . . . 5 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
5 elfz0lmr 13712 . . . . . . . . 9 (𝐽 ∈ (0...(β™―β€˜πΉ)) β†’ (𝐽 = 0 ∨ 𝐽 ∈ (1..^(β™―β€˜πΉ)) ∨ 𝐽 = (β™―β€˜πΉ)))
6 elfzo1 13647 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ (1..^(β™―β€˜πΉ)) ↔ (𝐼 ∈ β„• ∧ (β™―β€˜πΉ) ∈ β„• ∧ 𝐼 < (β™―β€˜πΉ)))
7 nnnn0 12444 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„• β†’ (β™―β€˜πΉ) ∈ β„•0)
873ad2ant2 1134 . . . . . . . . . . . . . . . . . . . . 21 ((𝐼 ∈ β„• ∧ (β™―β€˜πΉ) ∈ β„• ∧ 𝐼 < (β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) ∈ β„•0)
96, 8sylbi 216 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ (β™―β€˜πΉ) ∈ β„•0)
109adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (β™―β€˜πΉ) ∈ β„•0)
11 fvinim0ffz 13716 . . . . . . . . . . . . . . . . . . 19 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (β™―β€˜πΉ) ∈ β„•0) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… ↔ ((π‘ƒβ€˜0) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))))))
1210, 11sylan2 593 . . . . . . . . . . . . . . . . . 18 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… ↔ ((π‘ƒβ€˜0) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))))))
13 fveq2 6862 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐽 = 0 β†’ (π‘ƒβ€˜π½) = (π‘ƒβ€˜0))
1413eqeq2d 2742 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐽 = 0 β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) ↔ (π‘ƒβ€˜πΌ) = (π‘ƒβ€˜0)))
1514ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) ↔ (π‘ƒβ€˜πΌ) = (π‘ƒβ€˜0)))
16 ffun 6691 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ Fun 𝑃)
1716adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ Fun 𝑃)
18 fdm 6697 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ dom 𝑃 = (0...(β™―β€˜πΉ)))
19 fzo0ss1 13627 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (1..^(β™―β€˜πΉ)) βŠ† (0..^(β™―β€˜πΉ))
20 fzossfz 13616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0..^(β™―β€˜πΉ)) βŠ† (0...(β™―β€˜πΉ))
2119, 20sstri 3971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (1..^(β™―β€˜πΉ)) βŠ† (0...(β™―β€˜πΉ))
2221sseli 3958 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ 𝐼 ∈ (0...(β™―β€˜πΉ)))
23 eleq2 2821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑃 = (0...(β™―β€˜πΉ)) β†’ (𝐼 ∈ dom 𝑃 ↔ 𝐼 ∈ (0...(β™―β€˜πΉ))))
2422, 23imbitrrid 245 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (dom 𝑃 = (0...(β™―β€˜πΉ)) β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ 𝐼 ∈ dom 𝑃))
2518, 24syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ 𝐼 ∈ dom 𝑃))
2625imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ 𝐼 ∈ dom 𝑃)
2717, 26jca 512 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
2827adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
29 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ 𝐼 ∈ (1..^(β™―β€˜πΉ)))
30 funfvima 7200 . . . . . . . . . . . . . . . . . . . . . . . 24 ((Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃) β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ (π‘ƒβ€˜πΌ) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
3128, 29, 30sylc 65 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (π‘ƒβ€˜πΌ) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ))))
32 eleq1 2820 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜0) β†’ ((π‘ƒβ€˜πΌ) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ↔ (π‘ƒβ€˜0) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
3331, 32syl5ibcom 244 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜0) β†’ (π‘ƒβ€˜0) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
3415, 33sylbid 239 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) β†’ (π‘ƒβ€˜0) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
35 nnel 3055 . . . . . . . . . . . . . . . . . . . . 21 (Β¬ (π‘ƒβ€˜0) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ↔ (π‘ƒβ€˜0) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ))))
3634, 35syl6ibr 251 . . . . . . . . . . . . . . . . . . . 20 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) β†’ Β¬ (π‘ƒβ€˜0) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
3736necon2ad 2954 . . . . . . . . . . . . . . . . . . 19 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜0) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
3837adantrd 492 . . . . . . . . . . . . . . . . . 18 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (((π‘ƒβ€˜0) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
3912, 38sylbid 239 . . . . . . . . . . . . . . . . 17 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
4039ex 413 . . . . . . . . . . . . . . . 16 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))))
4140com23 86 . . . . . . . . . . . . . . 15 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))))
4241a1d 25 . . . . . . . . . . . . . 14 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
43423imp 1111 . . . . . . . . . . . . 13 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
4443com12 32 . . . . . . . . . . . 12 ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
4544a1d 25 . . . . . . . . . . 11 ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (𝐼 β‰  𝐽 β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))))
4645ex 413 . . . . . . . . . 10 (𝐽 = 0 β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ (𝐼 β‰  𝐽 β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
47 fvres 6881 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜πΌ) = (π‘ƒβ€˜πΌ))
4847adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜πΌ) = (π‘ƒβ€˜πΌ))
4948adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜πΌ) = (π‘ƒβ€˜πΌ))
5049eqcomd 2737 . . . . . . . . . . . . . . . . 17 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (π‘ƒβ€˜πΌ) = ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜πΌ))
51 fvres 6881 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (1..^(β™―β€˜πΉ)) β†’ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜π½) = (π‘ƒβ€˜π½))
5251ad2antrl 726 . . . . . . . . . . . . . . . . . 18 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜π½) = (π‘ƒβ€˜π½))
5352eqcomd 2737 . . . . . . . . . . . . . . . . 17 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (π‘ƒβ€˜π½) = ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜π½))
5450, 53eqeq12d 2747 . . . . . . . . . . . . . . . 16 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) ↔ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜πΌ) = ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜π½)))
55 fssres 6728 . . . . . . . . . . . . . . . . . . . 20 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (1..^(β™―β€˜πΉ)) βŠ† (0...(β™―β€˜πΉ))) β†’ (𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
5621, 55mpan2 689 . . . . . . . . . . . . . . . . . . 19 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ (𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
57 df-f1 6521 . . . . . . . . . . . . . . . . . . . 20 ((𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ) ↔ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ)))))
5857biimpri 227 . . . . . . . . . . . . . . . . . . 19 (((𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ)))) β†’ (𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ))
5956, 58sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ)))) β†’ (𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ))
60593adant3 1132 . . . . . . . . . . . . . . . . 17 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ))
61 simpr 485 . . . . . . . . . . . . . . . . . 18 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))))
6261ancomd 462 . . . . . . . . . . . . . . . . 17 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (1..^(β™―β€˜πΉ))))
63 f1veqaeq 7224 . . . . . . . . . . . . . . . . 17 (((𝑃 β†Ύ (1..^(β™―β€˜πΉ))):(1..^(β™―β€˜πΉ))–1-1β†’(Vtxβ€˜πΊ) ∧ (𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (1..^(β™―β€˜πΉ)))) β†’ (((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜πΌ) = ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜π½) β†’ 𝐼 = 𝐽))
6460, 62, 63syl2an2r 683 . . . . . . . . . . . . . . . 16 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜πΌ) = ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜π½) β†’ 𝐼 = 𝐽))
6554, 64sylbid 239 . . . . . . . . . . . . . . 15 (((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ (𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) β†’ 𝐼 = 𝐽))
6665ancoms 459 . . . . . . . . . . . . . 14 (((𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) ∧ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…)) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) β†’ 𝐼 = 𝐽))
6766necon3d 2960 . . . . . . . . . . . . 13 (((𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) ∧ (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…)) β†’ (𝐼 β‰  𝐽 β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
6867ex 413 . . . . . . . . . . . 12 ((𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (𝐼 β‰  𝐽 β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))))
6968com23 86 . . . . . . . . . . 11 ((𝐽 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (𝐼 β‰  𝐽 β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))))
7069ex 413 . . . . . . . . . 10 (𝐽 ∈ (1..^(β™―β€˜πΉ)) β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ (𝐼 β‰  𝐽 β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
719adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (β™―β€˜πΉ) ∈ β„•0)
7271, 11sylan2 593 . . . . . . . . . . . . . . . . . 18 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… ↔ ((π‘ƒβ€˜0) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))))))
73 fveq2 6862 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐽 = (β™―β€˜πΉ) β†’ (π‘ƒβ€˜π½) = (π‘ƒβ€˜(β™―β€˜πΉ)))
7473eqeq2d 2742 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐽 = (β™―β€˜πΉ) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) ↔ (π‘ƒβ€˜πΌ) = (π‘ƒβ€˜(β™―β€˜πΉ))))
7574ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) ↔ (π‘ƒβ€˜πΌ) = (π‘ƒβ€˜(β™―β€˜πΉ))))
7627adantrl 714 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
77 simprr 771 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ 𝐼 ∈ (1..^(β™―β€˜πΉ)))
7876, 77, 30sylc 65 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (π‘ƒβ€˜πΌ) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ))))
79 eleq1 2820 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜(β™―β€˜πΉ)) β†’ ((π‘ƒβ€˜πΌ) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ↔ (π‘ƒβ€˜(β™―β€˜πΉ)) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
8078, 79syl5ibcom 244 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜(β™―β€˜πΉ)) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
8175, 80sylbid 239 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
82 nnel 3055 . . . . . . . . . . . . . . . . . . . . 21 (Β¬ (π‘ƒβ€˜(β™―β€˜πΉ)) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ↔ (π‘ƒβ€˜(β™―β€˜πΉ)) ∈ (𝑃 β€œ (1..^(β™―β€˜πΉ))))
8381, 82syl6ibr 251 . . . . . . . . . . . . . . . . . . . 20 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) β†’ Β¬ (π‘ƒβ€˜(β™―β€˜πΉ)) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ)))))
8483necon2ad 2954 . . . . . . . . . . . . . . . . . . 19 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ ((π‘ƒβ€˜(β™―β€˜πΉ)) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
8584adantld 491 . . . . . . . . . . . . . . . . . 18 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (((π‘ƒβ€˜0) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ))) ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) βˆ‰ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
8672, 85sylbid 239 . . . . . . . . . . . . . . . . 17 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ (𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ)))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
8786ex 413 . . . . . . . . . . . . . . . 16 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ ((𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))))
8887com23 86 . . . . . . . . . . . . . . 15 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ ((𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))))
8988a1d 25 . . . . . . . . . . . . . 14 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ ((𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
90893imp 1111 . . . . . . . . . . . . 13 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ ((𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
9190com12 32 . . . . . . . . . . . 12 ((𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
9291a1d 25 . . . . . . . . . . 11 ((𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (𝐼 β‰  𝐽 β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))))
9392ex 413 . . . . . . . . . 10 (𝐽 = (β™―β€˜πΉ) β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ (𝐼 β‰  𝐽 β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
9446, 70, 933jaoi 1427 . . . . . . . . 9 ((𝐽 = 0 ∨ 𝐽 ∈ (1..^(β™―β€˜πΉ)) ∨ 𝐽 = (β™―β€˜πΉ)) β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ (𝐼 β‰  𝐽 β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
955, 94syl 17 . . . . . . . 8 (𝐽 ∈ (0...(β™―β€˜πΉ)) β†’ (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ (𝐼 β‰  𝐽 β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
96953imp21 1114 . . . . . . 7 ((𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (0...(β™―β€˜πΉ)) ∧ 𝐼 β‰  𝐽) β†’ ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
9796com12 32 . . . . . 6 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ ((𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (0...(β™―β€˜πΉ)) ∧ 𝐼 β‰  𝐽) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
98973exp 1119 . . . . 5 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ ((𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (0...(β™―β€˜πΉ)) ∧ 𝐼 β‰  𝐽) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
992, 4, 983syl 18 . . . 4 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ ((𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (0...(β™―β€˜πΉ)) ∧ 𝐼 β‰  𝐽) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))))
100993imp 1111 . . 3 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ ((𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (0...(β™―β€˜πΉ)) ∧ 𝐼 β‰  𝐽) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
1011, 100sylbi 216 . 2 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ ((𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (0...(β™―β€˜πΉ)) ∧ 𝐼 β‰  𝐽) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½)))
102101imp 407 1 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ (𝐼 ∈ (1..^(β™―β€˜πΉ)) ∧ 𝐽 ∈ (0...(β™―β€˜πΉ)) ∧ 𝐼 β‰  𝐽)) β†’ (π‘ƒβ€˜πΌ) β‰  (π‘ƒβ€˜π½))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2939   βˆ‰ wnel 3045   ∩ cin 3927   βŠ† wss 3928  βˆ…c0 4302  {cpr 4608   class class class wbr 5125  β—‘ccnv 5652  dom cdm 5653   β†Ύ cres 5655   β€œ cima 5656  Fun wfun 6510  βŸΆwf 6512  β€“1-1β†’wf1 6513  β€˜cfv 6516  (class class class)co 7377  0cc0 11075  1c1 11076   < clt 11213  β„•cn 12177  β„•0cn0 12437  ...cfz 13449  ..^cfzo 13592  β™―chash 14255  Vtxcvtx 28044  Walkscwlks 28641  Trailsctrls 28735  Pathscpths 28757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-1st 7941  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8670  df-map 8789  df-en 8906  df-dom 8907  df-sdom 8908  df-fin 8909  df-card 9899  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-n0 12438  df-z 12524  df-uz 12788  df-fz 13450  df-fzo 13593  df-hash 14256  df-word 14430  df-wlks 28644  df-trls 28737  df-pths 28761
This theorem is referenced by:  pthdadjvtx  28775  upgr4cycl4dv4e  29226
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