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Mirrors > Home > MPE Home > Th. List > df-pths | Structured version Visualization version GIF version |
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 28103). 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.) |
Ref | Expression |
---|---|
df-pths | ⊢ Paths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cpths 28076 | . 2 class Paths | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3431 | . . 3 class V | |
4 | vf | . . . . . . 7 setvar 𝑓 | |
5 | 4 | cv 1541 | . . . . . 6 class 𝑓 |
6 | vp | . . . . . . 7 setvar 𝑝 | |
7 | 6 | cv 1541 | . . . . . 6 class 𝑝 |
8 | 2 | cv 1541 | . . . . . . 7 class 𝑔 |
9 | ctrls 28055 | . . . . . . 7 class Trails | |
10 | 8, 9 | cfv 6432 | . . . . . 6 class (Trails‘𝑔) |
11 | 5, 7, 10 | wbr 5079 | . . . . 5 wff 𝑓(Trails‘𝑔)𝑝 |
12 | c1 10873 | . . . . . . . . 9 class 1 | |
13 | chash 14042 | . . . . . . . . . 10 class ♯ | |
14 | 5, 13 | cfv 6432 | . . . . . . . . 9 class (♯‘𝑓) |
15 | cfzo 13381 | . . . . . . . . 9 class ..^ | |
16 | 12, 14, 15 | co 7271 | . . . . . . . 8 class (1..^(♯‘𝑓)) |
17 | 7, 16 | cres 5592 | . . . . . . 7 class (𝑝 ↾ (1..^(♯‘𝑓))) |
18 | 17 | ccnv 5589 | . . . . . 6 class ◡(𝑝 ↾ (1..^(♯‘𝑓))) |
19 | 18 | wfun 6426 | . . . . 5 wff Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) |
20 | cc0 10872 | . . . . . . . . 9 class 0 | |
21 | 20, 14 | cpr 4569 | . . . . . . . 8 class {0, (♯‘𝑓)} |
22 | 7, 21 | cima 5593 | . . . . . . 7 class (𝑝 “ {0, (♯‘𝑓)}) |
23 | 7, 16 | cima 5593 | . . . . . . 7 class (𝑝 “ (1..^(♯‘𝑓))) |
24 | 22, 23 | cin 3891 | . . . . . 6 class ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) |
25 | c0 4262 | . . . . . 6 class ∅ | |
26 | 24, 25 | wceq 1542 | . . . . 5 wff ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅ |
27 | 11, 19, 26 | w3a 1086 | . . . 4 wff (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅) |
28 | 27, 4, 6 | copab 5141 | . . 3 class {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)} |
29 | 2, 3, 28 | cmpt 5162 | . 2 class (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}) |
30 | 1, 29 | wceq 1542 | 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 28084 pthsfval 28085 |
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