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Theorem usgr2pthlem 28753
Description: Lemma for usgr2pth 28754. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)
Hypotheses
Ref Expression
usgr2pthlem.v 𝑉 = (Vtxβ€˜πΊ)
usgr2pthlem.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
usgr2pthlem ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
Distinct variable groups:   𝑖,𝐹   π‘₯,𝐹,𝑦,𝑧   π‘₯,𝐺,𝑦,𝑧   𝑖,𝐼   π‘₯,𝐼,𝑦,𝑧   𝑃,𝑖   π‘₯,𝑃,𝑦,𝑧   π‘₯,𝑉,𝑦,𝑧
Allowed substitution hints:   𝐺(𝑖)   𝑉(𝑖)

Proof of Theorem usgr2pthlem
StepHypRef Expression
1 0nn0 12433 . . . . . . . . . . . . . 14 0 ∈ β„•0
2 2nn0 12435 . . . . . . . . . . . . . 14 2 ∈ β„•0
3 0le2 12260 . . . . . . . . . . . . . 14 0 ≀ 2
4 elfz2nn0 13538 . . . . . . . . . . . . . 14 (0 ∈ (0...2) ↔ (0 ∈ β„•0 ∧ 2 ∈ β„•0 ∧ 0 ≀ 2))
51, 2, 3, 4mpbir3an 1342 . . . . . . . . . . . . 13 0 ∈ (0...2)
6 ffvelcdm 7033 . . . . . . . . . . . . 13 ((𝑃:(0...2)βŸΆπ‘‰ ∧ 0 ∈ (0...2)) β†’ (π‘ƒβ€˜0) ∈ 𝑉)
75, 6mpan2 690 . . . . . . . . . . . 12 (𝑃:(0...2)βŸΆπ‘‰ β†’ (π‘ƒβ€˜0) ∈ 𝑉)
87adantl 483 . . . . . . . . . . 11 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜0) ∈ 𝑉)
9 1nn0 12434 . . . . . . . . . . . . . . . . . 18 1 ∈ β„•0
10 1le2 12367 . . . . . . . . . . . . . . . . . 18 1 ≀ 2
11 elfz2nn0 13538 . . . . . . . . . . . . . . . . . 18 (1 ∈ (0...2) ↔ (1 ∈ β„•0 ∧ 2 ∈ β„•0 ∧ 1 ≀ 2))
129, 2, 10, 11mpbir3an 1342 . . . . . . . . . . . . . . . . 17 1 ∈ (0...2)
13 ffvelcdm 7033 . . . . . . . . . . . . . . . . 17 ((𝑃:(0...2)βŸΆπ‘‰ ∧ 1 ∈ (0...2)) β†’ (π‘ƒβ€˜1) ∈ 𝑉)
1412, 13mpan2 690 . . . . . . . . . . . . . . . 16 (𝑃:(0...2)βŸΆπ‘‰ β†’ (π‘ƒβ€˜1) ∈ 𝑉)
1514adantl 483 . . . . . . . . . . . . . . 15 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜1) ∈ 𝑉)
16 simpr 486 . . . . . . . . . . . . . . . . . . 19 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ 𝐺 ∈ USGraph)
17 fvex 6856 . . . . . . . . . . . . . . . . . . 19 (π‘ƒβ€˜1) ∈ V
1816, 17jctir 522 . . . . . . . . . . . . . . . . . 18 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ (𝐺 ∈ USGraph ∧ (π‘ƒβ€˜1) ∈ V))
19 prcom 4694 . . . . . . . . . . . . . . . . . . . . . 22 {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} = {(π‘ƒβ€˜1), (π‘ƒβ€˜0)}
2019eqeq2i 2746 . . . . . . . . . . . . . . . . . . . . 21 ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ↔ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜0)})
2120biimpi 215 . . . . . . . . . . . . . . . . . . . 20 ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β†’ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜0)})
2221adantr 482 . . . . . . . . . . . . . . . . . . 19 (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜0)})
2322ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜0)})
24 usgr2pthlem.i . . . . . . . . . . . . . . . . . . 19 𝐼 = (iEdgβ€˜πΊ)
2524usgrnloopv 28190 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜1) ∈ V) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜0)} β†’ (π‘ƒβ€˜1) β‰  (π‘ƒβ€˜0)))
2618, 23, 25sylc 65 . . . . . . . . . . . . . . . . 17 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ (π‘ƒβ€˜1) β‰  (π‘ƒβ€˜0))
2726adantr 482 . . . . . . . . . . . . . . . 16 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜1) β‰  (π‘ƒβ€˜0))
2817elsn 4602 . . . . . . . . . . . . . . . . 17 ((π‘ƒβ€˜1) ∈ {(π‘ƒβ€˜0)} ↔ (π‘ƒβ€˜1) = (π‘ƒβ€˜0))
2928necon3bbii 2988 . . . . . . . . . . . . . . . 16 (Β¬ (π‘ƒβ€˜1) ∈ {(π‘ƒβ€˜0)} ↔ (π‘ƒβ€˜1) β‰  (π‘ƒβ€˜0))
3027, 29sylibr 233 . . . . . . . . . . . . . . 15 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ Β¬ (π‘ƒβ€˜1) ∈ {(π‘ƒβ€˜0)})
3115, 30eldifd 3922 . . . . . . . . . . . . . 14 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜1) ∈ (𝑉 βˆ– {(π‘ƒβ€˜0)}))
3231adantr 482 . . . . . . . . . . . . 13 (((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) β†’ (π‘ƒβ€˜1) ∈ (𝑉 βˆ– {(π‘ƒβ€˜0)}))
33 sneq 4597 . . . . . . . . . . . . . . . 16 (π‘₯ = (π‘ƒβ€˜0) β†’ {π‘₯} = {(π‘ƒβ€˜0)})
3433difeq2d 4083 . . . . . . . . . . . . . . 15 (π‘₯ = (π‘ƒβ€˜0) β†’ (𝑉 βˆ– {π‘₯}) = (𝑉 βˆ– {(π‘ƒβ€˜0)}))
3534eleq2d 2820 . . . . . . . . . . . . . 14 (π‘₯ = (π‘ƒβ€˜0) β†’ ((π‘ƒβ€˜1) ∈ (𝑉 βˆ– {π‘₯}) ↔ (π‘ƒβ€˜1) ∈ (𝑉 βˆ– {(π‘ƒβ€˜0)})))
3635adantl 483 . . . . . . . . . . . . 13 (((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) β†’ ((π‘ƒβ€˜1) ∈ (𝑉 βˆ– {π‘₯}) ↔ (π‘ƒβ€˜1) ∈ (𝑉 βˆ– {(π‘ƒβ€˜0)})))
3732, 36mpbird 257 . . . . . . . . . . . 12 (((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) β†’ (π‘ƒβ€˜1) ∈ (𝑉 βˆ– {π‘₯}))
38 2re 12232 . . . . . . . . . . . . . . . . . . . 20 2 ∈ ℝ
3938leidi 11694 . . . . . . . . . . . . . . . . . . 19 2 ≀ 2
40 elfz2nn0 13538 . . . . . . . . . . . . . . . . . . 19 (2 ∈ (0...2) ↔ (2 ∈ β„•0 ∧ 2 ∈ β„•0 ∧ 2 ≀ 2))
412, 2, 39, 40mpbir3an 1342 . . . . . . . . . . . . . . . . . 18 2 ∈ (0...2)
42 ffvelcdm 7033 . . . . . . . . . . . . . . . . . 18 ((𝑃:(0...2)βŸΆπ‘‰ ∧ 2 ∈ (0...2)) β†’ (π‘ƒβ€˜2) ∈ 𝑉)
4341, 42mpan2 690 . . . . . . . . . . . . . . . . 17 (𝑃:(0...2)βŸΆπ‘‰ β†’ (π‘ƒβ€˜2) ∈ 𝑉)
4443adantl 483 . . . . . . . . . . . . . . . 16 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜2) ∈ 𝑉)
4524usgrf1 28165 . . . . . . . . . . . . . . . . . . . . . 22 (𝐺 ∈ USGraph β†’ 𝐼:dom 𝐼–1-1β†’ran 𝐼)
4645ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ 𝐼:dom 𝐼–1-1β†’ran 𝐼)
47 simpl 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) β†’ 𝐹:(0..^2)–1-1β†’dom 𝐼)
4847ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ 𝐹:(0..^2)–1-1β†’dom 𝐼)
4946, 48jca 513 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (𝐼:dom 𝐼–1-1β†’ran 𝐼 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼))
50 2nn 12231 . . . . . . . . . . . . . . . . . . . . . . 23 2 ∈ β„•
51 lbfzo0 13618 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ∈ (0..^2) ↔ 2 ∈ β„•)
5250, 51mpbir 230 . . . . . . . . . . . . . . . . . . . . . 22 0 ∈ (0..^2)
53 1lt2 12329 . . . . . . . . . . . . . . . . . . . . . . 23 1 < 2
54 elfzo0 13619 . . . . . . . . . . . . . . . . . . . . . . 23 (1 ∈ (0..^2) ↔ (1 ∈ β„•0 ∧ 2 ∈ β„• ∧ 1 < 2))
559, 50, 53, 54mpbir3an 1342 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ (0..^2)
5652, 55pm3.2i 472 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ (0..^2) ∧ 1 ∈ (0..^2))
5756a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2)))
58 0ne1 12229 . . . . . . . . . . . . . . . . . . . . 21 0 β‰  1
5958a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ 0 β‰  1)
6049, 57, 593jca 1129 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ ((𝐼:dom 𝐼–1-1β†’ran 𝐼 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) ∧ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2)) ∧ 0 β‰  1))
61 simpr 486 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
6261ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
63 2f1fvneq 7208 . . . . . . . . . . . . . . . . . . 19 (((𝐼:dom 𝐼–1-1β†’ran 𝐼 ∧ 𝐹:(0..^2)–1-1β†’dom 𝐼) ∧ (0 ∈ (0..^2) ∧ 1 ∈ (0..^2)) ∧ 0 β‰  1) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β‰  {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
6460, 62, 63sylc 65 . . . . . . . . . . . . . . . . . 18 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β‰  {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})
65 necom 2994 . . . . . . . . . . . . . . . . . . 19 ((π‘ƒβ€˜2) β‰  (π‘ƒβ€˜0) ↔ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2))
66 fvex 6856 . . . . . . . . . . . . . . . . . . . . 21 (π‘ƒβ€˜0) ∈ V
67 fvex 6856 . . . . . . . . . . . . . . . . . . . . 21 (π‘ƒβ€˜2) ∈ V
6866, 17, 673pm3.2i 1340 . . . . . . . . . . . . . . . . . . . 20 ((π‘ƒβ€˜0) ∈ V ∧ (π‘ƒβ€˜1) ∈ V ∧ (π‘ƒβ€˜2) ∈ V)
69 fvexd 6858 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ (π‘ƒβ€˜0) ∈ V)
70 simpl 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
7170ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
7216, 69, 71jca31 516 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ ((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜0) ∈ V) ∧ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
7372adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ ((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜0) ∈ V) ∧ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
7424usgrnloopv 28190 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜0) ∈ V) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜1)))
7574imp 408 . . . . . . . . . . . . . . . . . . . . 21 (((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜0) ∈ V) ∧ (πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜1))
7673, 75syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜1))
77 pr1nebg 4816 . . . . . . . . . . . . . . . . . . . 20 ((((π‘ƒβ€˜0) ∈ V ∧ (π‘ƒβ€˜1) ∈ V ∧ (π‘ƒβ€˜2) ∈ V) ∧ (π‘ƒβ€˜0) β‰  (π‘ƒβ€˜1)) β†’ ((π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2) ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β‰  {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
7868, 76, 77sylancr 588 . . . . . . . . . . . . . . . . . . 19 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ ((π‘ƒβ€˜0) β‰  (π‘ƒβ€˜2) ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β‰  {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
7965, 78bitrid 283 . . . . . . . . . . . . . . . . . 18 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ ((π‘ƒβ€˜2) β‰  (π‘ƒβ€˜0) ↔ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} β‰  {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
8064, 79mpbird 257 . . . . . . . . . . . . . . . . 17 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜2) β‰  (π‘ƒβ€˜0))
81 fvexd 6858 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ (π‘ƒβ€˜2) ∈ V)
82 prcom 4694 . . . . . . . . . . . . . . . . . . . . . . . 24 {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)}
8382eqeq2i 2746 . . . . . . . . . . . . . . . . . . . . . . 23 ((πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} ↔ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)})
8483biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 ((πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} β†’ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)})
8584adantl 483 . . . . . . . . . . . . . . . . . . . . 21 (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)})
8685ad2antlr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)})
8716, 81, 86jca31 516 . . . . . . . . . . . . . . . . . . 19 (((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) β†’ ((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜2) ∈ V) ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)}))
8887adantr 482 . . . . . . . . . . . . . . . . . 18 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ ((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜2) ∈ V) ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)}))
8924usgrnloopv 28190 . . . . . . . . . . . . . . . . . . 19 ((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜2) ∈ V) β†’ ((πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)} β†’ (π‘ƒβ€˜2) β‰  (π‘ƒβ€˜1)))
9089imp 408 . . . . . . . . . . . . . . . . . 18 (((𝐺 ∈ USGraph ∧ (π‘ƒβ€˜2) ∈ V) ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜2), (π‘ƒβ€˜1)}) β†’ (π‘ƒβ€˜2) β‰  (π‘ƒβ€˜1))
9188, 90syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜2) β‰  (π‘ƒβ€˜1))
9280, 91nelprd 4618 . . . . . . . . . . . . . . . 16 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ Β¬ (π‘ƒβ€˜2) ∈ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
9344, 92eldifd 3922 . . . . . . . . . . . . . . 15 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (π‘ƒβ€˜2) ∈ (𝑉 βˆ– {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
9493ad2antrr 725 . . . . . . . . . . . . . 14 ((((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) ∧ 𝑦 = (π‘ƒβ€˜1)) β†’ (π‘ƒβ€˜2) ∈ (𝑉 βˆ– {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
95 preq12 4697 . . . . . . . . . . . . . . . . 17 ((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) β†’ {π‘₯, 𝑦} = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})
9695difeq2d 4083 . . . . . . . . . . . . . . . 16 ((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) β†’ (𝑉 βˆ– {π‘₯, 𝑦}) = (𝑉 βˆ– {(π‘ƒβ€˜0), (π‘ƒβ€˜1)}))
9796eleq2d 2820 . . . . . . . . . . . . . . 15 ((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) β†’ ((π‘ƒβ€˜2) ∈ (𝑉 βˆ– {π‘₯, 𝑦}) ↔ (π‘ƒβ€˜2) ∈ (𝑉 βˆ– {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})))
9897adantll 713 . . . . . . . . . . . . . 14 ((((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) ∧ 𝑦 = (π‘ƒβ€˜1)) β†’ ((π‘ƒβ€˜2) ∈ (𝑉 βˆ– {π‘₯, 𝑦}) ↔ (π‘ƒβ€˜2) ∈ (𝑉 βˆ– {(π‘ƒβ€˜0), (π‘ƒβ€˜1)})))
9994, 98mpbird 257 . . . . . . . . . . . . 13 ((((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) ∧ 𝑦 = (π‘ƒβ€˜1)) β†’ (π‘ƒβ€˜2) ∈ (𝑉 βˆ– {π‘₯, 𝑦}))
100 eqcom 2740 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ = (π‘ƒβ€˜0) ↔ (π‘ƒβ€˜0) = π‘₯)
101 eqcom 2740 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = (π‘ƒβ€˜1) ↔ (π‘ƒβ€˜1) = 𝑦)
102 eqcom 2740 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (π‘ƒβ€˜2) ↔ (π‘ƒβ€˜2) = 𝑧)
103100, 101, 1023anbi123i 1156 . . . . . . . . . . . . . . . . . . 19 ((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1) ∧ 𝑧 = (π‘ƒβ€˜2)) ↔ ((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧))
104103biimpi 215 . . . . . . . . . . . . . . . . . 18 ((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ ((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧))
105104ad4ant123 1173 . . . . . . . . . . . . . . . . 17 ((((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) β†’ ((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧))
106100biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ = (π‘ƒβ€˜0) β†’ (π‘ƒβ€˜0) = π‘₯)
107106ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ (π‘ƒβ€˜0) = π‘₯)
108101biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (π‘ƒβ€˜1) β†’ (π‘ƒβ€˜1) = 𝑦)
109108ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ (π‘ƒβ€˜1) = 𝑦)
110107, 109preq12d 4703 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} = {π‘₯, 𝑦})
111110eqeq2d 2744 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ↔ (πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦}))
112102biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 = (π‘ƒβ€˜2) β†’ (π‘ƒβ€˜2) = 𝑧)
113112adantl 483 . . . . . . . . . . . . . . . . . . . . 21 (((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ (π‘ƒβ€˜2) = 𝑧)
114109, 113preq12d 4703 . . . . . . . . . . . . . . . . . . . 20 (((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} = {𝑦, 𝑧})
115114eqeq2d 2744 . . . . . . . . . . . . . . . . . . 19 (((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ ((πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)} ↔ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))
116111, 115anbi12d 632 . . . . . . . . . . . . . . . . . 18 (((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) ↔ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))
117116biimpa 478 . . . . . . . . . . . . . . . . 17 ((((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) β†’ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))
118105, 117jca 513 . . . . . . . . . . . . . . . 16 ((((π‘₯ = (π‘ƒβ€˜0) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) β†’ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))
119118exp41 436 . . . . . . . . . . . . . . 15 (π‘₯ = (π‘ƒβ€˜0) β†’ (𝑦 = (π‘ƒβ€˜1) β†’ (𝑧 = (π‘ƒβ€˜2) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
120119adantl 483 . . . . . . . . . . . . . 14 (((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) β†’ (𝑦 = (π‘ƒβ€˜1) β†’ (𝑧 = (π‘ƒβ€˜2) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
121120imp31 419 . . . . . . . . . . . . 13 (((((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) ∧ 𝑦 = (π‘ƒβ€˜1)) ∧ 𝑧 = (π‘ƒβ€˜2)) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
12299, 121rspcimedv 3571 . . . . . . . . . . . 12 ((((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) ∧ 𝑦 = (π‘ƒβ€˜1)) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
12337, 122rspcimedv 3571 . . . . . . . . . . 11 (((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) ∧ π‘₯ = (π‘ƒβ€˜0)) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
1248, 123rspcimedv 3571 . . . . . . . . . 10 ((((𝐹:(0..^2)–1-1β†’dom 𝐼 ∧ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})) ∧ 𝐺 ∈ USGraph) ∧ 𝑃:(0...2)βŸΆπ‘‰) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
125124exp41 436 . . . . . . . . 9 (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (𝐺 ∈ USGraph β†’ (𝑃:(0...2)βŸΆπ‘‰ β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))))
126125com15 101 . . . . . . . 8 (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (𝐺 ∈ USGraph β†’ (𝑃:(0...2)βŸΆπ‘‰ β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))))
127126pm2.43i 52 . . . . . . 7 (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (𝐺 ∈ USGraph β†’ (𝑃:(0...2)βŸΆπ‘‰ β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
128127com12 32 . . . . . 6 (𝐺 ∈ USGraph β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (𝑃:(0...2)βŸΆπ‘‰ β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
129128adantr 482 . . . . 5 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}) β†’ (𝑃:(0...2)βŸΆπ‘‰ β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
130 oveq2 7366 . . . . . . . 8 ((β™―β€˜πΉ) = 2 β†’ (0..^(β™―β€˜πΉ)) = (0..^2))
131130raleqdv 3312 . . . . . . 7 ((β™―β€˜πΉ) = 2 β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ βˆ€π‘– ∈ (0..^2)(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
132 fzo0to2pr 13663 . . . . . . . . 9 (0..^2) = {0, 1}
133132raleqi 3310 . . . . . . . 8 (βˆ€π‘– ∈ (0..^2)(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ βˆ€π‘– ∈ {0, 1} (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})
134 2wlklem 28657 . . . . . . . 8 (βˆ€π‘– ∈ {0, 1} (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
135133, 134bitri 275 . . . . . . 7 (βˆ€π‘– ∈ (0..^2)(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)}))
136131, 135bitrdi 287 . . . . . 6 ((β™―β€˜πΉ) = 2 β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})))
137136adantl 483 . . . . 5 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ ((πΌβ€˜(πΉβ€˜0)) = {(π‘ƒβ€˜0), (π‘ƒβ€˜1)} ∧ (πΌβ€˜(πΉβ€˜1)) = {(π‘ƒβ€˜1), (π‘ƒβ€˜2)})))
138 oveq2 7366 . . . . . . . 8 ((β™―β€˜πΉ) = 2 β†’ (0...(β™―β€˜πΉ)) = (0...2))
139138feq2d 6655 . . . . . . 7 ((β™―β€˜πΉ) = 2 β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ↔ 𝑃:(0...2)βŸΆπ‘‰))
140139adantl 483 . . . . . 6 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ↔ 𝑃:(0...2)βŸΆπ‘‰))
141 f1eq2 6735 . . . . . . . . 9 ((0..^(β™―β€˜πΉ)) = (0..^2) β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ↔ 𝐹:(0..^2)–1-1β†’dom 𝐼))
142130, 141syl 17 . . . . . . . 8 ((β™―β€˜πΉ) = 2 β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ↔ 𝐹:(0..^2)–1-1β†’dom 𝐼))
143142imbi1d 342 . . . . . . 7 ((β™―β€˜πΉ) = 2 β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ↔ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
144143adantl 483 . . . . . 6 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))) ↔ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))))
145140, 144imbi12d 345 . . . . 5 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ ((𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧})))) ↔ (𝑃:(0...2)βŸΆπ‘‰ β†’ (𝐹:(0..^2)–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
146129, 137, 1453imtr4d 294 . . . 4 ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
147146com14 96 . . 3 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
148147com23 86 . 2 (𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 β†’ (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))))
1491483imp 1112 1 ((𝐹:(0..^(β™―β€˜πΉ))–1-1β†’dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) β†’ ((𝐺 ∈ USGraph ∧ (β™―β€˜πΉ) = 2) β†’ βˆƒπ‘₯ ∈ 𝑉 βˆƒπ‘¦ ∈ (𝑉 βˆ– {π‘₯})βˆƒπ‘§ ∈ (𝑉 βˆ– {π‘₯, 𝑦})(((π‘ƒβ€˜0) = π‘₯ ∧ (π‘ƒβ€˜1) = 𝑦 ∧ (π‘ƒβ€˜2) = 𝑧) ∧ ((πΌβ€˜(πΉβ€˜0)) = {π‘₯, 𝑦} ∧ (πΌβ€˜(πΉβ€˜1)) = {𝑦, 𝑧}))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3444   βˆ– cdif 3908  {csn 4587  {cpr 4589   class class class wbr 5106  dom cdm 5634  ran crn 5635  βŸΆwf 6493  β€“1-1β†’wf1 6494  β€˜cfv 6497  (class class class)co 7358  0cc0 11056  1c1 11057   + caddc 11059   < clt 11194   ≀ cle 11195  β„•cn 12158  2c2 12213  β„•0cn0 12418  ...cfz 13430  ..^cfzo 13573  β™―chash 14236  Vtxcvtx 27989  iEdgciedg 27990  USGraphcusgr 28142
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-oadd 8417  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-dju 9842  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-n0 12419  df-z 12505  df-uz 12769  df-fz 13431  df-fzo 13574  df-hash 14237  df-umgr 28076  df-usgr 28144
This theorem is referenced by:  usgr2pth  28754
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