 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-crct Structured version   Visualization version   GIF version

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
StepHypRef 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
65cv 1474 . . . . . 6 class 𝑓
7 vp . . . . . . 7 setvar 𝑝
87cv 1474 . . . . . 6 class 𝑝
92cv 1474 . . . . . . 7 class 𝑣
103cv 1474 . . . . . . 7 class 𝑒
11 ctrail 26027 . . . . . . 7 class Trails
129, 10, 11co 6549 . . . . . 6 class (𝑣 Trails 𝑒)
136, 8, 12wbr 4583 . . . . 5 wff 𝑓(𝑣 Trails 𝑒)𝑝
14 cc0 9815 . . . . . . 7 class 0
1514, 8cfv 5804 . . . . . 6 class (𝑝‘0)
16 chash 12979 . . . . . . . 8 class #
176, 16cfv 5804 . . . . . . 7 class (#‘𝑓)
1817, 8cfv 5804 . . . . . 6 class (𝑝‘(#‘𝑓))
1915, 18wceq 1475 . . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓))
2013, 19wa 383 . . . 4 wff (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))
2120, 5, 7copab 4642 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}
222, 3, 4, 4, 21cmpt2 6551 . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
231, 22wceq 1475 1 wff Circuits = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})
 Colors of variables: wff setvar class This definition is referenced by:  crcts  26150  crctistrl  26156
 Copyright terms: Public domain W3C validator