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Definition df-pths 26814
 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 upgrwlkdvspth 26837). 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 26810 . 2 class Paths
2 vg . . 3 setvar 𝑔
3 cvv 3332 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1623 . . . . . 6 class 𝑓
6 vp . . . . . . 7 setvar 𝑝
76cv 1623 . . . . . 6 class 𝑝
82cv 1623 . . . . . . 7 class 𝑔
9 ctrls 26789 . . . . . . 7 class Trails
108, 9cfv 6041 . . . . . 6 class (Trails‘𝑔)
115, 7, 10wbr 4796 . . . . 5 wff 𝑓(Trails‘𝑔)𝑝
12 c1 10121 . . . . . . . . 9 class 1
13 chash 13303 . . . . . . . . . 10 class
145, 13cfv 6041 . . . . . . . . 9 class (♯‘𝑓)
15 cfzo 12651 . . . . . . . . 9 class ..^
1612, 14, 15co 6805 . . . . . . . 8 class (1..^(♯‘𝑓))
177, 16cres 5260 . . . . . . 7 class (𝑝 ↾ (1..^(♯‘𝑓)))
1817ccnv 5257 . . . . . 6 class (𝑝 ↾ (1..^(♯‘𝑓)))
1918wfun 6035 . . . . 5 wff Fun (𝑝 ↾ (1..^(♯‘𝑓)))
20 cc0 10120 . . . . . . . . 9 class 0
2120, 14cpr 4315 . . . . . . . 8 class {0, (♯‘𝑓)}
227, 21cima 5261 . . . . . . 7 class (𝑝 “ {0, (♯‘𝑓)})
237, 16cima 5261 . . . . . . 7 class (𝑝 “ (1..^(♯‘𝑓)))
2422, 23cin 3706 . . . . . 6 class ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓))))
25 c0 4050 . . . . . 6 class
2624, 25wceq 1624 . . . . 5 wff ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅
2711, 19, 26w3a 1072 . . . 4 wff (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)
2827, 4, 6copab 4856 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}
292, 3, 28cmpt 4873 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
301, 29wceq 1624 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:  relpths  26818  pthsfval  26819
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