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

Definition df-wlkson 26706
Description: Define the collection of walks with particular endpoints (in a hypergraph). The predicate 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 can be read as "The pair 𝐹, 𝑃 represents a walk from vertex 𝐴 to vertex 𝐵 in a graph 𝐺", see also iswlkon 26763. This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)
Assertion
Ref Expression
df-wlkson WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
Distinct variable groups:   𝑓,𝑔,𝑝   𝑎,𝑏,𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-wlkson
StepHypRef Expression
1 cwlkson 26703 . 2 class WalksOn
2 vg . . 3 setvar 𝑔
3 cvv 3340 . . 3 class V
4 va . . . 4 setvar 𝑎
5 vb . . . 4 setvar 𝑏
62cv 1631 . . . . 5 class 𝑔
7 cvtx 26073 . . . . 5 class Vtx
86, 7cfv 6049 . . . 4 class (Vtx‘𝑔)
9 vf . . . . . . . 8 setvar 𝑓
109cv 1631 . . . . . . 7 class 𝑓
11 vp . . . . . . . 8 setvar 𝑝
1211cv 1631 . . . . . . 7 class 𝑝
13 cwlks 26702 . . . . . . . 8 class Walks
146, 13cfv 6049 . . . . . . 7 class (Walks‘𝑔)
1510, 12, 14wbr 4804 . . . . . 6 wff 𝑓(Walks‘𝑔)𝑝
16 cc0 10128 . . . . . . . 8 class 0
1716, 12cfv 6049 . . . . . . 7 class (𝑝‘0)
184cv 1631 . . . . . . 7 class 𝑎
1917, 18wceq 1632 . . . . . 6 wff (𝑝‘0) = 𝑎
20 chash 13311 . . . . . . . . 9 class
2110, 20cfv 6049 . . . . . . . 8 class (♯‘𝑓)
2221, 12cfv 6049 . . . . . . 7 class (𝑝‘(♯‘𝑓))
235cv 1631 . . . . . . 7 class 𝑏
2422, 23wceq 1632 . . . . . 6 wff (𝑝‘(♯‘𝑓)) = 𝑏
2515, 19, 24w3a 1072 . . . . 5 wff (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)
2625, 9, 11copab 4864 . . . 4 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}
274, 5, 8, 8, 26cmpt2 6815 . . 3 class (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)})
282, 3, 27cmpt 4881 . 2 class (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
291, 28wceq 1632 1 wff WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(♯‘𝑓)) = 𝑏)}))
Colors of variables: wff setvar class
This definition is referenced by:  wlkson  26762  wlkonprop  26764
  Copyright terms: Public domain W3C validator