Theorem clwwlknonfin

Description: In a finite graph G , the set of closed walks on vertex X of length N is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018) (Revised by AV, 25-Feb-2022) (Proof shortened by AV, 24-Mar-2022)

	Ref	Expression
Hypothesis	clwwlknonfin.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	clwwlknonfin	⊢ ( 𝑉 ∈ Fin → ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		clwwlknonfin.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		clwwlknon	⊢ ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) = { 𝑤 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑋 }
3	1	eleq1i	⊢ ( 𝑉 ∈ Fin ↔ ( Vtx ‘ 𝐺 ) ∈ Fin )
4		clwwlknfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 ClWWalksN 𝐺 ) ∈ Fin )
5	3 4	sylbi	⊢ ( 𝑉 ∈ Fin → ( 𝑁 ClWWalksN 𝐺 ) ∈ Fin )
6		rabfi	⊢ ( ( 𝑁 ClWWalksN 𝐺 ) ∈ Fin → { 𝑤 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ∈ Fin )
7	5 6	syl	⊢ ( 𝑉 ∈ Fin → { 𝑤 ∈ ( 𝑁 ClWWalksN 𝐺 ) ∣ ( 𝑤 ‘ 0 ) = 𝑋 } ∈ Fin )
8	2 7	eqeltrid	⊢ ( 𝑉 ∈ Fin → ( 𝑋 ( ClWWalksNOn ‘ 𝐺 ) 𝑁 ) ∈ Fin )