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Definition df-pthson 40925
 Description: Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.)
Assertion
Ref Expression
df-pthson PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(PathS‘𝑔)𝑝)}))
Distinct variable groups:   𝑓,𝑔,𝑝   𝑎,𝑏,𝑔,𝑓,𝑝

Detailed syntax breakdown of Definition df-pthson
StepHypRef Expression
1 cpthson 40921 . 2 class PathsOn
2 vg . . 3 setvar 𝑔
3 cvv 3173 . . 3 class V
4 va . . . 4 setvar 𝑎
5 vb . . . 4 setvar 𝑏
62cv 1474 . . . . 5 class 𝑔
7 cvtx 25673 . . . . 5 class Vtx
86, 7cfv 5804 . . . 4 class (Vtx‘𝑔)
9 vf . . . . . . . 8 setvar 𝑓
109cv 1474 . . . . . . 7 class 𝑓
11 vp . . . . . . . 8 setvar 𝑝
1211cv 1474 . . . . . . 7 class 𝑝
134cv 1474 . . . . . . . 8 class 𝑎
145cv 1474 . . . . . . . 8 class 𝑏
15 ctrlson 40900 . . . . . . . . 9 class TrailsOn
166, 15cfv 5804 . . . . . . . 8 class (TrailsOn‘𝑔)
1713, 14, 16co 6549 . . . . . . 7 class (𝑎(TrailsOn‘𝑔)𝑏)
1810, 12, 17wbr 4583 . . . . . 6 wff 𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝
19 cpths 40919 . . . . . . . 8 class PathS
206, 19cfv 5804 . . . . . . 7 class (PathS‘𝑔)
2110, 12, 20wbr 4583 . . . . . 6 wff 𝑓(PathS‘𝑔)𝑝
2218, 21wa 383 . . . . 5 wff (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(PathS‘𝑔)𝑝)
2322, 9, 11copab 4642 . . . 4 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(PathS‘𝑔)𝑝)}
244, 5, 8, 8, 23cmpt2 6551 . . 3 class (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(PathS‘𝑔)𝑝)})
252, 3, 24cmpt 4643 . 2 class (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(PathS‘𝑔)𝑝)}))
261, 25wceq 1475 1 wff PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(PathS‘𝑔)𝑝)}))
 Colors of variables: wff setvar class This definition is referenced by:  pthsonfval  40946  pthsonprop  40950
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