Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-pths Structured version   Visualization version   GIF version

Definition df-pths 40921
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.) (Revised by AV, 9-Jan-2021.)

Assertion
Ref Expression
df-pths PathS = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-pths
StepHypRef Expression
1 cpths 40917 . 2 class PathS
2 vg . . 3 setvar 𝑔
3 cvv 3168 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1473 . . . . . 6 class 𝑓
6 vp . . . . . . 7 setvar 𝑝
76cv 1473 . . . . . 6 class 𝑝
82cv 1473 . . . . . . 7 class 𝑔
9 ctrls 40897 . . . . . . 7 class TrailS
108, 9cfv 5786 . . . . . 6 class (TrailS‘𝑔)
115, 7, 10wbr 4573 . . . . 5 wff 𝑓(TrailS‘𝑔)𝑝
12 c1 9789 . . . . . . . . 9 class 1
13 chash 12930 . . . . . . . . . 10 class #
145, 13cfv 5786 . . . . . . . . 9 class (#‘𝑓)
15 cfzo 12285 . . . . . . . . 9 class ..^
1612, 14, 15co 6523 . . . . . . . 8 class (1..^(#‘𝑓))
177, 16cres 5026 . . . . . . 7 class (𝑝 ↾ (1..^(#‘𝑓)))
1817ccnv 5023 . . . . . 6 class (𝑝 ↾ (1..^(#‘𝑓)))
1918wfun 5780 . . . . 5 wff Fun (𝑝 ↾ (1..^(#‘𝑓)))
20 cc0 9788 . . . . . . . . 9 class 0
2120, 14cpr 4122 . . . . . . . 8 class {0, (#‘𝑓)}
227, 21cima 5027 . . . . . . 7 class (𝑝 “ {0, (#‘𝑓)})
237, 16cima 5027 . . . . . . 7 class (𝑝 “ (1..^(#‘𝑓)))
2422, 23cin 3534 . . . . . 6 class ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓))))
25 c0 3869 . . . . . 6 class
2624, 25wceq 1474 . . . . 5 wff ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅
2711, 19, 26w3a 1030 . . . 4 wff (𝑓(TrailS‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)
2827, 4, 6copab 4632 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}
292, 3, 28cmpt 4633 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
301, 29wceq 1474 1 wff PathS = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
Colors of variables: wff setvar class
This definition is referenced by:  pthsfval  40925
  Copyright terms: Public domain W3C validator