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Database GRAPH THEORY Walks, paths and cycles Walks as words wspthnonfi
Description: In a finite graph, the set of simple paths of a fixed length between two
vertices is finite. (Contributed by Alexander van der Vekens , 4-Mar-2018) (Revised by AV , 15-May-2021)
Ref
Expression
Assertion
wspthnonfi ⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WSPathsNOn 𝐺 ) 𝐵 ) ∈ Fin )
Proof
Step
Hyp
Ref
Expression
1
wwlksnonfi ⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) ∈ Fin )
2
wspthsswwlknon ⊢ ( 𝐴 ( 𝑁 WSPathsNOn 𝐺 ) 𝐵 ) ⊆ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 )
3
2
a1i ⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WSPathsNOn 𝐺 ) 𝐵 ) ⊆ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) )
4
1 3
ssfid ⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WSPathsNOn 𝐺 ) 𝐵 ) ∈ Fin )