Theorem clwwlknfi

Description: If there is only a finite number of vertices, the number of closed walks of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018) (Revised by AV, 25-Apr-2021) (Proof shortened by AV, 22-Mar-2022) (Proof shortened by JJ, 18-Nov-2022)

	Ref	Expression
Assertion	clwwlknfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 ClWWalksN 𝐺 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		clwwlkn	⊢ ( 𝑁 ClWWalksN 𝐺 ) = { 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 𝑁 }
2		wrdnfi	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } ∈ Fin )
3		clwwlksswrd	⊢ ( ClWWalks ‘ 𝐺 ) ⊆ Word ( Vtx ‘ 𝐺 )
4		rabss2	⊢ ( ( ClWWalks ‘ 𝐺 ) ⊆ Word ( Vtx ‘ 𝐺 ) → { 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } ⊆ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } )
5	3 4	mp1i	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } ⊆ { 𝑤 ∈ Word ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } )
6	2 5	ssfid	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → { 𝑤 ∈ ( ClWWalks ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑤 ) = 𝑁 } ∈ Fin )
7	1 6	eqeltrid	⊢ ( ( Vtx ‘ 𝐺 ) ∈ Fin → ( 𝑁 ClWWalksN 𝐺 ) ∈ Fin )