Definition df-pth 26038

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 26138).

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

Step	Hyp	Ref	Expression
1		cpath 26028	. 2 class Paths
2		vv	. . 3 setvar 𝑣
3		ve	. . 3 setvar 𝑒
4		cvv 3173	. . 3 class V
5		vf	. . . . . . 7 setvar 𝑓
6	5	cv 1474	. . . . . 6 class 𝑓
7		vp	. . . . . . 7 setvar 𝑝
8	7	cv 1474	. . . . . 6 class 𝑝
9	2	cv 1474	. . . . . . 7 class 𝑣
10	3	cv 1474	. . . . . . 7 class 𝑒
11		ctrail 26027	. . . . . . 7 class Trails
12	9, 10, 11	co 6549	. . . . . 6 class (𝑣 Trails 𝑒)
13	6, 8, 12	wbr 4583	. . . . 5 wff 𝑓(𝑣 Trails 𝑒)𝑝
14		c1 9816	. . . . . . . . 9 class 1
15		chash 12979	. . . . . . . . . 10 class #
16	6, 15	cfv 5804	. . . . . . . . 9 class (#‘𝑓)
17		cfzo 12334	. . . . . . . . 9 class ..^
18	14, 16, 17	co 6549	. . . . . . . 8 class (1..^(#‘𝑓))
19	8, 18	cres 5040	. . . . . . 7 class (𝑝 ↾ (1..^(#‘𝑓)))
20	19	ccnv 5037	. . . . . 6 class ◡(𝑝 ↾ (1..^(#‘𝑓)))
21	20	wfun 5798	. . . . 5 wff Fun ◡(𝑝 ↾ (1..^(#‘𝑓)))
22		cc0 9815	. . . . . . . . 9 class 0
23	22, 16	cpr 4127	. . . . . . . 8 class {0, (#‘𝑓)}
24	8, 23	cima 5041	. . . . . . 7 class (𝑝 “ {0, (#‘𝑓)})
25	8, 18	cima 5041	. . . . . . 7 class (𝑝 “ (1..^(#‘𝑓)))
26	24, 25	cin 3539	. . . . . 6 class ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓))))
27		c0 3874	. . . . . 6 class ∅
28	26, 27	wceq 1475	. . . . 5 wff ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅
29	13, 21, 28	w3a 1031	. . . 4 wff (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)
30	29, 5, 7	copab 4642	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}
31	2, 3, 4, 4, 30	cmpt2 6551	. 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})
32	1, 31	wceq 1475	1 wff Paths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})

This definition is referenced by:  pths  26096  pthistrl  26102  pthdepisspth  26104