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Definition df-wspthsnon 29126
Description: Define the collection of simple paths of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
Assertion
Ref Expression
df-wspthsnon WSPathsNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}))
Distinct variable group:   π‘Ž,𝑏,𝑓,𝑔,𝑛,𝑀

Detailed syntax breakdown of Definition df-wspthsnon
StepHypRef Expression
1 cwwspthsnon 29121 . 2 class WSPathsNOn
2 vn . . 3 setvar 𝑛
3 vg . . 3 setvar 𝑔
4 cn0 12474 . . 3 class β„•0
5 cvv 3474 . . 3 class V
6 va . . . 4 setvar π‘Ž
7 vb . . . 4 setvar 𝑏
83cv 1540 . . . . 5 class 𝑔
9 cvtx 28294 . . . . 5 class Vtx
108, 9cfv 6543 . . . 4 class (Vtxβ€˜π‘”)
11 vf . . . . . . . 8 setvar 𝑓
1211cv 1540 . . . . . . 7 class 𝑓
13 vw . . . . . . . 8 setvar 𝑀
1413cv 1540 . . . . . . 7 class 𝑀
156cv 1540 . . . . . . . 8 class π‘Ž
167cv 1540 . . . . . . . 8 class 𝑏
17 cspthson 29010 . . . . . . . . 9 class SPathsOn
188, 17cfv 6543 . . . . . . . 8 class (SPathsOnβ€˜π‘”)
1915, 16, 18co 7411 . . . . . . 7 class (π‘Ž(SPathsOnβ€˜π‘”)𝑏)
2012, 14, 19wbr 5148 . . . . . 6 wff 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀
2120, 11wex 1781 . . . . 5 wff βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀
222cv 1540 . . . . . . 7 class 𝑛
23 cwwlksnon 29119 . . . . . . 7 class WWalksNOn
2422, 8, 23co 7411 . . . . . 6 class (𝑛 WWalksNOn 𝑔)
2515, 16, 24co 7411 . . . . 5 class (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏)
2621, 13, 25crab 3432 . . . 4 class {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}
276, 7, 10, 10, 26cmpo 7413 . . 3 class (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀})
282, 3, 4, 5, 27cmpo 7413 . 2 class (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}))
291, 28wceq 1541 1 wff WSPathsNOn = (𝑛 ∈ β„•0, 𝑔 ∈ V ↦ (π‘Ž ∈ (Vtxβ€˜π‘”), 𝑏 ∈ (Vtxβ€˜π‘”) ↦ {𝑀 ∈ (π‘Ž(𝑛 WWalksNOn 𝑔)𝑏) ∣ βˆƒπ‘“ 𝑓(π‘Ž(SPathsOnβ€˜π‘”)𝑏)𝑀}))
Colors of variables: wff setvar class
This definition is referenced by:  wspthsnon  29144  iswspthsnon  29148  wspthnonp  29151
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