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

Definition df-spthson 26821
Description: Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 9-Jan-2021.)
Assertion
Ref Expression
df-spthson SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(SPaths‘𝑔)𝑝)}))
Distinct variable groups:   𝑓,𝑔,𝑝   𝑎,𝑏,𝑔,𝑓,𝑝

Detailed syntax breakdown of Definition df-spthson
StepHypRef Expression
1 cspthson 26817 . 2 class SPathsOn
2 vg . . 3 setvar 𝑔
3 cvv 3336 . . 3 class V
4 va . . . 4 setvar 𝑎
5 vb . . . 4 setvar 𝑏
62cv 1627 . . . . 5 class 𝑔
7 cvtx 26069 . . . . 5 class Vtx
86, 7cfv 6045 . . . 4 class (Vtx‘𝑔)
9 vf . . . . . . . 8 setvar 𝑓
109cv 1627 . . . . . . 7 class 𝑓
11 vp . . . . . . . 8 setvar 𝑝
1211cv 1627 . . . . . . 7 class 𝑝
134cv 1627 . . . . . . . 8 class 𝑎
145cv 1627 . . . . . . . 8 class 𝑏
15 ctrlson 26794 . . . . . . . . 9 class TrailsOn
166, 15cfv 6045 . . . . . . . 8 class (TrailsOn‘𝑔)
1713, 14, 16co 6809 . . . . . . 7 class (𝑎(TrailsOn‘𝑔)𝑏)
1810, 12, 17wbr 4800 . . . . . 6 wff 𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝
19 cspths 26815 . . . . . . . 8 class SPaths
206, 19cfv 6045 . . . . . . 7 class (SPaths‘𝑔)
2110, 12, 20wbr 4800 . . . . . 6 wff 𝑓(SPaths‘𝑔)𝑝
2218, 21wa 383 . . . . 5 wff (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(SPaths‘𝑔)𝑝)
2322, 9, 11copab 4860 . . . 4 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(SPaths‘𝑔)𝑝)}
244, 5, 8, 8, 23cmpt2 6811 . . 3 class (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(SPaths‘𝑔)𝑝)})
252, 3, 24cmpt 4877 . 2 class (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(SPaths‘𝑔)𝑝)}))
261, 25wceq 1628 1 wff SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(SPaths‘𝑔)𝑝)}))
Colors of variables: wff setvar class
This definition is referenced by:  spthson  26843  spthonprop  26847
  Copyright terms: Public domain W3C validator