Theorem stgrvtx0

Description: The center ("internal node") of a star graph S_N. (Contributed by AV, 12-Sep-2025)

	Ref	Expression
Hypotheses	stgrvtx0.g	⊢ 𝐺 = ( StarGr ‘ 𝑁 )
	stgrvtx0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	stgrvtx0	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ 𝑉 )

Proof

Step	Hyp	Ref	Expression
1		stgrvtx0.g	⊢ 𝐺 = ( StarGr ‘ 𝑁 )
2		stgrvtx0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
3		stgrvtx	⊢ ( 𝑁 ∈ ℕ0 → ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) )
4	1	fveq2i	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ ( StarGr ‘ 𝑁 ) )
5	2 4	eqtri	⊢ 𝑉 = ( Vtx ‘ ( StarGr ‘ 𝑁 ) )
6	5	eqeq1i	⊢ ( 𝑉 = ( 0 ... 𝑁 ) ↔ ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) )
7		0elfz	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) )
8		eleq2	⊢ ( 𝑉 = ( 0 ... 𝑁 ) → ( 0 ∈ 𝑉 ↔ 0 ∈ ( 0 ... 𝑁 ) ) )
9	7 8	syl5ibrcom	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑉 = ( 0 ... 𝑁 ) → 0 ∈ 𝑉 ) )
10	6 9	biimtrrid	⊢ ( 𝑁 ∈ ℕ0 → ( ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) → 0 ∈ 𝑉 ) )
11	3 10	mpd	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ 𝑉 )