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Mirrors  >  Home  >  MPE Home  >  Th. List  >  pthdivtx Structured version   Visualization version   GIF version
Theorem pthdivtx 29253
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 29247 . . 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 29221 . . . . 5 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
3 eqid 2730 . . . . . 6 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
43wlkp 29140 . . . . 5 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ))
5 elfz0lmr 13751 . . . . . . . . 9 (𝐽 ∈ (0...(β™―β€˜πΉ)) β†’ (𝐽 = 0 ∨ 𝐽 ∈ (1..^(β™―β€˜πΉ)) ∨ 𝐽 = (β™―β€˜πΉ)))
6 elfzo1 13686 . . . . . . . . . . . . . . . . . . . . 21 (𝐼 ∈ (1..^(β™―β€˜πΉ)) ↔ (𝐼 ∈ β„• ∧ (β™―β€˜πΉ) ∈ β„• ∧ 𝐼 < (β™―β€˜πΉ)))
7 nnnn0 12483 . . . . . . . . . . . . . . . . . . . . . 22 ((β™―β€˜πΉ) ∈ β„• β†’ (β™―β€˜πΉ) ∈ β„•0)
873ad2ant2 1132 . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 19 ((𝐽 = 0 ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (β™―β€˜πΉ) ∈ β„•0)
11 fvinim0ffz 13755 . . . . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐽 = 0 β†’ (π‘ƒβ€˜π½) = (π‘ƒβ€˜0))
1413eqeq2d 2741 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐽 = 0 β†’ ((π‘ƒβ€˜πΌ) = (π‘ƒβ€˜π½) ↔ (π‘ƒβ€˜πΌ) = (π‘ƒβ€˜0)))
1514ad2antrl 724 . . . . . . . . . . . . . . . . . . . . . 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 6719 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ Fun 𝑃)
1716adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ Fun 𝑃)
18 fdm 6725 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) β†’ dom 𝑃 = (0...(β™―β€˜πΉ)))
19 fzo0ss1 13666 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (1..^(β™―β€˜πΉ)) βŠ† (0..^(β™―β€˜πΉ))
20 fzossfz 13655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (0..^(β™―β€˜πΉ)) βŠ† (0...(β™―β€˜πΉ))
2119, 20sstri 3990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (1..^(β™―β€˜πΉ)) βŠ† (0...(β™―β€˜πΉ))
2221sseli 3977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ 𝐼 ∈ (0...(β™―β€˜πΉ)))
23 eleq2 2820 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ 𝐼 ∈ dom 𝑃)
2717, 26jca 510 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (Fun 𝑃 ∧ 𝐼 ∈ dom 𝑃))
2827adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . . . 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 7233 . . . . . . . . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . . . . . . . . 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 3054 . . . . . . . . . . . . . . . . . . . . 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, 35imbitrrdi 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 2953 . . . . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 1109 . . . . . . . . . . . . 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 411 . . . . . . . . . 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 6909 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ (1..^(β™―β€˜πΉ)) β†’ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜πΌ) = (π‘ƒβ€˜πΌ))
4847adantl 480 . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . 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 6909 . . . . . . . . . . . . . . . . . . 19 (𝐽 ∈ (1..^(β™―β€˜πΉ)) β†’ ((𝑃 β†Ύ (1..^(β™―β€˜πΉ)))β€˜π½) = (π‘ƒβ€˜π½))
5251ad2antrl 724 . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . 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 6756 . . . . . . . . . . . . . . . . . . . 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 687 . . . . . . . . . . . . . . . . . . 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 6547 . . . . . . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . . . . . . 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 1130 . . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . 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 460 . . . . . . . . . . . . . . . . 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 7258 . . . . . . . . . . . . . . . . 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 681 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 2959 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 411 . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 19 ((𝐽 = (β™―β€˜πΉ) ∧ 𝐼 ∈ (1..^(β™―β€˜πΉ))) β†’ (β™―β€˜πΉ) ∈ β„•0)
7271, 11sylan2 591 . . . . . . . . . . . . . . . . . 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 6890 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐽 = (β™―β€˜πΉ) β†’ (π‘ƒβ€˜π½) = (π‘ƒβ€˜(β™―β€˜πΉ)))
7473eqeq2d 2741 . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . . . 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 2819 . . . . . . . . . . . . . . . . . . . . . . 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 3054 . . . . . . . . . . . . . . . . . . . . 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, 82imbitrrdi 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 2953 . . . . . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 1109 . . . . . . . . . . . . 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 411 . . . . . . . . . 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 1425 . . . . . . . . 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 1112 . . . . . . 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 1117 . . . . 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 1109 . . 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 405 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 394   ∨ w3o 1084   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   βˆ‰ wnel 3044   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  {cpr 4629   class class class wbr 5147  β—‘ccnv 5674  dom cdm 5675   β†Ύ cres 5677   β€œ cima 5678  Fun wfun 6536  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   < clt 11252  β„•cn 12216  β„•0cn0 12476  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Vtxcvtx 28523  Walkscwlks 29120  Trailsctrls 29214  Pathscpths 29236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-wlks 29123  df-trls 29216  df-pths 29240
This theorem is referenced by:  pthdadjvtx  29254  upgr4cycl4dv4e  29705
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