Definition df-spth 26039

Description: Define the set of all Simple 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."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple 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, 20-Oct-2017.)

Assertion

Ref	Expression
df-spth	⊢ SPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡𝑝)})

Distinct variable group:   𝑣,𝑒,𝑓,𝑝

Detailed syntax breakdown of Definition df-spth

Step	Hyp	Ref	Expression
1		cspath 26029	. 2 class SPaths
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	8	ccnv 5037	. . . . . 6 class ◡𝑝
15	14	wfun 5798	. . . . 5 wff Fun ◡𝑝
16	13, 15	wa 383	. . . 4 wff (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡𝑝)
17	16, 5, 7	copab 4642	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡𝑝)}
18	2, 3, 4, 4, 17	cmpt2 6551	. 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡𝑝)})
19	1, 18	wceq 1475	1 wff SPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun ◡𝑝)})

This definition is referenced by:  spths  26097  spthispth  26103