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Metamath Proof Explorer

Theorem wwlksnonfi

Description: In a finite graph, the set of walks of a fixed length between two vertices is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018) (Revised by AV, 15-May-2021) (Proof shortened by AV, 15-Mar-2022)

Ref Expression
Assertion wwlksnonfi ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) ∈ Fin )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2 1 iswwlksnon ⊢ ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) = { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝐴 ∧ ( 𝑤 ‘ 𝑁 ) = 𝐵 ) }
3 wwlksnfi ⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 WWalksN 𝐺 ) ∈ Fin )
4 rabfi ⊢ ( ( 𝑁 WWalksN 𝐺 ) ∈ Fin → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝐴 ∧ ( 𝑤 ‘ 𝑁 ) = 𝐵 ) } ∈ Fin )
5 3 4 syl ⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ ( 𝑁 WWalksN 𝐺 ) ∣ ( ( 𝑤 ‘ 0 ) = 𝐴 ∧ ( 𝑤 ‘ 𝑁 ) = 𝐵 ) } ∈ Fin )
6 2 5 eqeltrid ⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝐴 ( 𝑁 WWalksNOn 𝐺 ) 𝐵 ) ∈ Fin )