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Definition df-wspthsnon 26629
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 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
Distinct variable group:   𝑎,𝑏,𝑓,𝑔,𝑛,𝑤

Detailed syntax breakdown of Definition df-wspthsnon
StepHypRef Expression
1 cwwspthsnon 26624 . 2 class WSPathsNOn
2 vn . . 3 setvar 𝑛
3 vg . . 3 setvar 𝑔
4 cn0 11252 . . 3 class 0
5 cvv 3190 . . 3 class V
6 va . . . 4 setvar 𝑎
7 vb . . . 4 setvar 𝑏
83cv 1479 . . . . 5 class 𝑔
9 cvtx 25808 . . . . 5 class Vtx
108, 9cfv 5857 . . . 4 class (Vtx‘𝑔)
11 vf . . . . . . . 8 setvar 𝑓
1211cv 1479 . . . . . . 7 class 𝑓
13 vw . . . . . . . 8 setvar 𝑤
1413cv 1479 . . . . . . 7 class 𝑤
156cv 1479 . . . . . . . 8 class 𝑎
167cv 1479 . . . . . . . 8 class 𝑏
17 cspthson 26514 . . . . . . . . 9 class SPathsOn
188, 17cfv 5857 . . . . . . . 8 class (SPathsOn‘𝑔)
1915, 16, 18co 6615 . . . . . . 7 class (𝑎(SPathsOn‘𝑔)𝑏)
2012, 14, 19wbr 4623 . . . . . 6 wff 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤
2120, 11wex 1701 . . . . 5 wff 𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤
222cv 1479 . . . . . . 7 class 𝑛
23 cwwlksnon 26622 . . . . . . 7 class WWalksNOn
2422, 8, 23co 6615 . . . . . 6 class (𝑛 WWalksNOn 𝑔)
2515, 16, 24co 6615 . . . . 5 class (𝑎(𝑛 WWalksNOn 𝑔)𝑏)
2621, 13, 25crab 2912 . . . 4 class {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}
276, 7, 10, 10, 26cmpt2 6617 . . 3 class (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤})
282, 3, 4, 5, 27cmpt2 6617 . 2 class (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
291, 28wceq 1480 1 wff WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
Colors of variables: wff setvar class
This definition is referenced by:  wspthsnon  26642  iswspthsnon  26644  wspthnonp  26647
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