Theorem clnbfiusgrfi

Description: The closed neighborhood of a vertex in a finite simple graph is a finite set. (Contributed by AV, 10-May-2025)

	Ref	Expression
Assertion	clnbfiusgrfi	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ClNeighbVtx 𝑁 ) ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		fusgrusgr	⊢ ( 𝐺 ∈ FinUSGraph → 𝐺 ∈ USGraph )
2	1	adantr	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐺 ∈ USGraph )
3		fusgrfis	⊢ ( 𝐺 ∈ FinUSGraph → ( Edg ‘ 𝐺 ) ∈ Fin )
4	3	adantr	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → ( Edg ‘ 𝐺 ) ∈ Fin )
5		simpr	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → 𝑁 ∈ ( Vtx ‘ 𝐺 ) )
6		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
7		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
8	6 7	clnbusgrfi	⊢ ( ( 𝐺 ∈ USGraph ∧ ( Edg ‘ 𝐺 ) ∈ Fin ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ClNeighbVtx 𝑁 ) ∈ Fin )
9	2 4 5 8	syl3anc	⊢ ( ( 𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ClNeighbVtx 𝑁 ) ∈ Fin )