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Theorem usgra2wlkspth 26149
 Description: In a undirected simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
Assertion
Ref Expression
usgra2wlkspth ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃))

Proof of Theorem usgra2wlkspth
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 wlkonprop 26063 . . . 4 (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))
2 simplr 788 . . . . . 6 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃)
3 iswlk 26048 . . . . . . . . . . . . . . . 16 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
433adant3 1074 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
5 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 ∈ Word dom 𝐸𝐹 ∈ Word dom 𝐸)
653ad2ant1 1075 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → 𝐹 ∈ Word dom 𝐸)
76ad4antlr 765 . . . . . . . . . . . . . . . . . . 19 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐹 ∈ Word dom 𝐸)
8 usgraf1 25889 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑉 USGrph 𝐸𝐸:dom 𝐸1-1→ran 𝐸)
983ad2ant1 1075 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → 𝐸:dom 𝐸1-1→ran 𝐸)
109adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐸:dom 𝐸1-1→ran 𝐸)
11 simp2 1055 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → (#‘𝐹) = 2)
1211adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (#‘𝐹) = 2)
137, 10, 123jca 1235 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝐹 ∈ Word dom 𝐸𝐸:dom 𝐸1-1→ran 𝐸 ∧ (#‘𝐹) = 2))
14 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝑃‘0) = 𝐴)
1514ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝑃‘0) = 𝐴)
16 simpr 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝑃‘(#‘𝐹)) = 𝐵)
1716ad3antlr 763 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝑃‘(#‘𝐹)) = 𝐵)
18 simp3 1056 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → 𝐴𝐵)
1918adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐴𝐵)
2015, 17, 193jca 1235 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵𝐴𝐵))
21 simp3 1056 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})
2221ad4antlr 765 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})
2320, 22jca 553 . . . . . . . . . . . . . . . . . . . 20 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵𝐴𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
24 usgra2wlkspthlem1 26147 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ Word dom 𝐸𝐸:dom 𝐸1-1→ran 𝐸 ∧ (#‘𝐹) = 2) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵𝐴𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → Fun 𝐹))
2513, 23, 24sylc 63 . . . . . . . . . . . . . . . . . . 19 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → Fun 𝐹)
267, 25jca 553 . . . . . . . . . . . . . . . . . 18 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹))
27 simp2 1055 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → 𝑃:(0...(#‘𝐹))⟶𝑉)
2827ad4antlr 765 . . . . . . . . . . . . . . . . . 18 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝑃:(0...(#‘𝐹))⟶𝑉)
2926, 28, 223jca 1235 . . . . . . . . . . . . . . . . 17 (((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
3029exp31 628 . . . . . . . . . . . . . . . 16 (((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))))
3130exp31 628 . . . . . . . . . . . . . . 15 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))))))
324, 31sylbid 229 . . . . . . . . . . . . . 14 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝑉 Walks 𝐸)𝑃 → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))))))
3332com13 86 . . . . . . . . . . . . 13 (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝐹(𝑉 Walks 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))))))
3433ex 449 . . . . . . . . . . . 12 ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (𝐹(𝑉 Walks 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))))))
3534com3r 85 . . . . . . . . . . 11 (𝐹(𝑉 Walks 𝐸)𝑃 → ((𝑃‘0) = 𝐴 → ((𝑃‘(#‘𝐹)) = 𝐵 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))))))
36353imp 1249 . . . . . . . . . 10 ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))))
3736impcom 445 . . . . . . . . 9 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))))
3837imp31 447 . . . . . . . 8 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
39 id 22 . . . . . . . . . . 11 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
40393adant3 1074 . . . . . . . . . 10 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
4140ad3antrrr 762 . . . . . . . . 9 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))
42 istrl 26067 . . . . . . . . 9 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
4341, 42syl 17 . . . . . . . 8 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
4438, 43mpbird 246 . . . . . . 7 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐹(𝑉 Trails 𝐸)𝑃)
45 2mwlk 26049 . . . . . . . . . . . . 13 (𝐹(𝑉 Walks 𝐸)𝑃 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉))
46 simpl 472 . . . . . . . . . . . . 13 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) → 𝐹 ∈ Word dom 𝐸)
4745, 46syl 17 . . . . . . . . . . . 12 (𝐹(𝑉 Walks 𝐸)𝑃𝐹 ∈ Word dom 𝐸)
48473ad2ant1 1075 . . . . . . . . . . 11 ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → 𝐹 ∈ Word dom 𝐸)
4948ad3antlr 763 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐹 ∈ Word dom 𝐸)
5011adantl 481 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (#‘𝐹) = 2)
5149, 50jca 553 . . . . . . . . 9 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2))
52 id 22 . . . . . . . . . . . 12 (𝑉 USGrph 𝐸𝑉 USGrph 𝐸)
53523ad2ant1 1075 . . . . . . . . . . 11 ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → 𝑉 USGrph 𝐸)
5453adantl 481 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝑉 USGrph 𝐸)
55 simpr 476 . . . . . . . . . . . . 13 ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃:(0...(#‘𝐹))⟶𝑉)
5645, 55syl 17 . . . . . . . . . . . 12 (𝐹(𝑉 Walks 𝐸)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
57563ad2ant1 1075 . . . . . . . . . . 11 ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → 𝑃:(0...(#‘𝐹))⟶𝑉)
5857ad3antlr 763 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝑃:(0...(#‘𝐹))⟶𝑉)
5954, 58jca 553 . . . . . . . . 9 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝑉 USGrph 𝐸𝑃:(0...(#‘𝐹))⟶𝑉))
6051, 59jca 553 . . . . . . . 8 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → ((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ (𝑉 USGrph 𝐸𝑃:(0...(#‘𝐹))⟶𝑉)))
61 simp2 1055 . . . . . . . . . . 11 ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝑃‘0) = 𝐴)
6261ad3antlr 763 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝑃‘0) = 𝐴)
63 simp3 1056 . . . . . . . . . . 11 ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (𝑃‘(#‘𝐹)) = 𝐵)
6463ad3antlr 763 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝑃‘(#‘𝐹)) = 𝐵)
6518adantl 481 . . . . . . . . . 10 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐴𝐵)
6662, 64, 653jca 1235 . . . . . . . . 9 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → ((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵𝐴𝐵))
674, 21syl6bi 242 . . . . . . . . . . . . 13 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝑉 Walks 𝐸)𝑃 → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
6867com12 32 . . . . . . . . . . . 12 (𝐹(𝑉 Walks 𝐸)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
69683ad2ant1 1075 . . . . . . . . . . 11 ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵) → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
7069impcom 445 . . . . . . . . . 10 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})
7170ad2antrr 758 . . . . . . . . 9 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})
7266, 71jca 553 . . . . . . . 8 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵𝐴𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
73 usgra2wlkspthlem2 26148 . . . . . . . 8 (((𝐹 ∈ Word dom 𝐸 ∧ (#‘𝐹) = 2) ∧ (𝑉 USGrph 𝐸𝑃:(0...(#‘𝐹))⟶𝑉)) → ((((𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵𝐴𝐵) ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) → Fun 𝑃))
7460, 72, 73sylc 63 . . . . . . 7 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → Fun 𝑃)
75 isspth 26099 . . . . . . . 8 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))
7641, 75syl 17 . . . . . . 7 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))
7744, 74, 76mpbir2and 959 . . . . . 6 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐹(𝑉 SPaths 𝐸)𝑃)
78 isspthon 26113 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)))
7978ad3antrrr 762 . . . . . 6 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)))
802, 77, 79mpbir2and 959 . . . . 5 ((((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) ∧ 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃) ∧ (𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵)) → 𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃)
8180exp31 628 . . . 4 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → 𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃)))
821, 81mpcom 37 . . 3 (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → 𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃))
8382com12 32 . 2 ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃))
84 spthonprp 26115 . . 3 (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)))
85 simpl 472 . . . 4 ((𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃)
8685adantl 481 . . 3 ((((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 SPaths 𝐸)𝑃)) → 𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃)
8784, 86syl 17 . 2 (𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃)
8883, 87impbid1 214 1 ((𝑉 USGrph 𝐸 ∧ (#‘𝐹) = 2 ∧ 𝐴𝐵) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  {cpr 4127   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038  ran crn 5039  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   USGrph cusg 25859   Walks cwalk 26026   Trails ctrail 26027   SPaths cspath 26029   WalkOn cwlkon 26030   SPathOn cspthon 26033 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-spth 26039  df-wlkon 26042  df-spthon 26045 This theorem is referenced by:  2pthwlkonot  26412
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