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Definition df-pth 25800
Description: Define the set of all Paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see wlkdvspth 25900).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Assertion
Ref Expression
df-pth Paths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
Distinct variable group:   𝑣,𝑒,𝑓,𝑝

Detailed syntax breakdown of Definition df-pth
StepHypRef Expression
1 cpath 25790 . 2 class Paths
2 vv . . 3 setvar 𝑣
3 ve . . 3 setvar 𝑒
4 cvv 3168 . . 3 class V
5 vf . . . . . . 7 setvar 𝑓
65cv 1473 . . . . . 6 class 𝑓
7 vp . . . . . . 7 setvar 𝑝
87cv 1473 . . . . . 6 class 𝑝
92cv 1473 . . . . . . 7 class 𝑣
103cv 1473 . . . . . . 7 class 𝑒
11 ctrail 25789 . . . . . . 7 class Trails
129, 10, 11co 6523 . . . . . 6 class (𝑣 Trails 𝑒)
136, 8, 12wbr 4573 . . . . 5 wff 𝑓(𝑣 Trails 𝑒)𝑝
14 c1 9789 . . . . . . . . 9 class 1
15 chash 12930 . . . . . . . . . 10 class #
166, 15cfv 5786 . . . . . . . . 9 class (#‘𝑓)
17 cfzo 12285 . . . . . . . . 9 class ..^
1814, 16, 17co 6523 . . . . . . . 8 class (1..^(#‘𝑓))
198, 18cres 5026 . . . . . . 7 class (𝑝 ↾ (1..^(#‘𝑓)))
2019ccnv 5023 . . . . . 6 class (𝑝 ↾ (1..^(#‘𝑓)))
2120wfun 5780 . . . . 5 wff Fun (𝑝 ↾ (1..^(#‘𝑓)))
22 cc0 9788 . . . . . . . . 9 class 0
2322, 16cpr 4122 . . . . . . . 8 class {0, (#‘𝑓)}
248, 23cima 5027 . . . . . . 7 class (𝑝 “ {0, (#‘𝑓)})
258, 18cima 5027 . . . . . . 7 class (𝑝 “ (1..^(#‘𝑓)))
2624, 25cin 3534 . . . . . 6 class ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓))))
27 c0 3869 . . . . . 6 class
2826, 27wceq 1474 . . . . 5 wff ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅
2913, 21, 28w3a 1030 . . . 4 wff (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)
3029, 5, 7copab 4632 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}
312, 3, 4, 4, 30cmpt2 6525 . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
321, 31wceq 1474 1 wff Paths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
Colors of variables: wff setvar class
This definition is referenced by:  pths  25858  pthistrl  25864  pthdepisspth  25866
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