Definition df-pths 28970

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 28993).

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

Step	Hyp	Ref	Expression
1		cpths 28966	. 2 class Paths
2		vg	. . 3 setvar 𝑔
3		cvv 3474	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1540	. . . . . 6 class 𝑓
6		vp	. . . . . . 7 setvar 𝑝
7	6	cv 1540	. . . . . 6 class 𝑝
8	2	cv 1540	. . . . . . 7 class 𝑔
9		ctrls 28944	. . . . . . 7 class Trails
10	8, 9	cfv 6543	. . . . . 6 class (Trails‘𝑔)
11	5, 7, 10	wbr 5148	. . . . 5 wff 𝑓(Trails‘𝑔)𝑝
12		c1 11110	. . . . . . . . 9 class 1
13		chash 14289	. . . . . . . . . 10 class ♯
14	5, 13	cfv 6543	. . . . . . . . 9 class (♯‘𝑓)
15		cfzo 13626	. . . . . . . . 9 class ..^
16	12, 14, 15	co 7408	. . . . . . . 8 class (1..^(♯‘𝑓))
17	7, 16	cres 5678	. . . . . . 7 class (𝑝 ↾ (1..^(♯‘𝑓)))
18	17	ccnv 5675	. . . . . 6 class ◡(𝑝 ↾ (1..^(♯‘𝑓)))
19	18	wfun 6537	. . . . 5 wff Fun ◡(𝑝 ↾ (1..^(♯‘𝑓)))
20		cc0 11109	. . . . . . . . 9 class 0
21	20, 14	cpr 4630	. . . . . . . 8 class {0, (♯‘𝑓)}
22	7, 21	cima 5679	. . . . . . 7 class (𝑝 “ {0, (♯‘𝑓)})
23	7, 16	cima 5679	. . . . . . 7 class (𝑝 “ (1..^(♯‘𝑓)))
24	22, 23	cin 3947	. . . . . 6 class ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓))))
25		c0 4322	. . . . . 6 class ∅
26	24, 25	wceq 1541	. . . . 5 wff ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅
27	11, 19, 26	w3a 1087	. . . 4 wff (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)
28	27, 4, 6	copab 5210	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)}
29	2, 3, 28	cmpt 5231	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
30	1, 29	wceq 1541	1 wff Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})

This definition is referenced by:  relpths  28974  pthsfval  28975