Definition df-crct 26040

Description: Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit 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)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Assertion

Ref	Expression
df-crct	⊢ Circuits = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Distinct variable group:   𝑣,𝑒,𝑓,𝑝

Detailed syntax breakdown of Definition df-crct

Step	Hyp	Ref	Expression
1		ccrct 26034	. 2 class Circuits
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		cc0 9815	. . . . . . 7 class 0
15	14, 8	cfv 5804	. . . . . 6 class (𝑝‘0)
16		chash 12979	. . . . . . . 8 class #
17	6, 16	cfv 5804	. . . . . . 7 class (#‘𝑓)
18	17, 8	cfv 5804	. . . . . 6 class (𝑝‘(#‘𝑓))
19	15, 18	wceq 1475	. . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓))
20	13, 19	wa 383	. . . 4 wff (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))
21	20, 5, 7	copab 4642	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}
22	2, 3, 4, 4, 21	cmpt2 6551	. 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
23	1, 22	wceq 1475	1 wff Circuits = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

This definition is referenced by:  crcts  26150  crctistrl  26156