Theorem stgrnbgr0

Description: All vertices of a star graph S_N except the center are in the (open) neighborhood of the center. (Contributed by AV, 12-Sep-2025)

	Ref	Expression
Hypotheses	stgrvtx0.g	⊢ 𝐺 = ( StarGr ‘ 𝑁 )
	stgrvtx0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	stgrnbgr0	⊢ ( 𝑁 ∈ ℕ0 → ( 𝐺 NeighbVtx 0 ) = ( 𝑉 ∖ { 0 } ) )

Proof

Step	Hyp	Ref	Expression
1		stgrvtx0.g	⊢ 𝐺 = ( StarGr ‘ 𝑁 )
2		stgrvtx0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
3	1 2	stgrvtx0	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ 𝑉 )
4		eqid	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 )
5	2 4	dfnbgr2	⊢ ( 0 ∈ 𝑉 → ( 𝐺 NeighbVtx 0 ) = { 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) ( 0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒 ) } )
6	3 5	syl	⊢ ( 𝑁 ∈ ℕ0 → ( 𝐺 NeighbVtx 0 ) = { 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) ( 0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒 ) } )
7		eleq2	⊢ ( 𝑒 = { 0 , 𝑥 } → ( 0 ∈ 𝑒 ↔ 0 ∈ { 0 , 𝑥 } ) )
8		eleq2	⊢ ( 𝑒 = { 0 , 𝑥 } → ( 𝑥 ∈ 𝑒 ↔ 𝑥 ∈ { 0 , 𝑥 } ) )
9	7 8	anbi12d	⊢ ( 𝑒 = { 0 , 𝑥 } → ( ( 0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒 ) ↔ ( 0 ∈ { 0 , 𝑥 } ∧ 𝑥 ∈ { 0 , 𝑥 } ) ) )
10		0elfz	⊢ ( 𝑁 ∈ ℕ0 → 0 ∈ ( 0 ... 𝑁 ) )
11	10	adantr	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → 0 ∈ ( 0 ... 𝑁 ) )
12		fz1ssfz0	⊢ ( 1 ... 𝑁 ) ⊆ ( 0 ... 𝑁 )
13	1	fveq2i	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ ( StarGr ‘ 𝑁 ) )
14	2 13	eqtri	⊢ 𝑉 = ( Vtx ‘ ( StarGr ‘ 𝑁 ) )
15		stgrvtx	⊢ ( 𝑁 ∈ ℕ0 → ( Vtx ‘ ( StarGr ‘ 𝑁 ) ) = ( 0 ... 𝑁 ) )
16	14 15	eqtrid	⊢ ( 𝑁 ∈ ℕ0 → 𝑉 = ( 0 ... 𝑁 ) )
17	16	difeq1d	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑉 ∖ { 0 } ) = ( ( 0 ... 𝑁 ) ∖ { 0 } ) )
18		fz0dif1	⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 𝑁 ) ∖ { 0 } ) = ( 1 ... 𝑁 ) )
19	18	eqimssd	⊢ ( 𝑁 ∈ ℕ0 → ( ( 0 ... 𝑁 ) ∖ { 0 } ) ⊆ ( 1 ... 𝑁 ) )
20	17 19	eqsstrd	⊢ ( 𝑁 ∈ ℕ0 → ( 𝑉 ∖ { 0 } ) ⊆ ( 1 ... 𝑁 ) )
21	20	sselda	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → 𝑥 ∈ ( 1 ... 𝑁 ) )
22	12 21	sselid	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → 𝑥 ∈ ( 0 ... 𝑁 ) )
23	11 22	prssd	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → { 0 , 𝑥 } ⊆ ( 0 ... 𝑁 ) )
24		preq2	⊢ ( 𝑛 = 𝑥 → { 0 , 𝑛 } = { 0 , 𝑥 } )
25	24	eqeq2d	⊢ ( 𝑛 = 𝑥 → ( { 0 , 𝑥 } = { 0 , 𝑛 } ↔ { 0 , 𝑥 } = { 0 , 𝑥 } ) )
26		eqidd	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → { 0 , 𝑥 } = { 0 , 𝑥 } )
27	25 21 26	rspcedvdw	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → ∃ 𝑛 ∈ ( 1 ... 𝑁 ) { 0 , 𝑥 } = { 0 , 𝑛 } )
28	1	fveq2i	⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ ( StarGr ‘ 𝑁 ) )
29	28	eleq2i	⊢ ( { 0 , 𝑥 } ∈ ( Edg ‘ 𝐺 ) ↔ { 0 , 𝑥 } ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) )
30		stgredgel	⊢ ( 𝑁 ∈ ℕ0 → ( { 0 , 𝑥 } ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( { 0 , 𝑥 } ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑛 ∈ ( 1 ... 𝑁 ) { 0 , 𝑥 } = { 0 , 𝑛 } ) ) )
31	30	adantr	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → ( { 0 , 𝑥 } ∈ ( Edg ‘ ( StarGr ‘ 𝑁 ) ) ↔ ( { 0 , 𝑥 } ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑛 ∈ ( 1 ... 𝑁 ) { 0 , 𝑥 } = { 0 , 𝑛 } ) ) )
32	29 31	bitrid	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → ( { 0 , 𝑥 } ∈ ( Edg ‘ 𝐺 ) ↔ ( { 0 , 𝑥 } ⊆ ( 0 ... 𝑁 ) ∧ ∃ 𝑛 ∈ ( 1 ... 𝑁 ) { 0 , 𝑥 } = { 0 , 𝑛 } ) ) )
33	23 27 32	mpbir2and	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → { 0 , 𝑥 } ∈ ( Edg ‘ 𝐺 ) )
34		prid2g	⊢ ( 𝑥 ∈ ( 𝑉 ∖ { 0 } ) → 𝑥 ∈ { 0 , 𝑥 } )
35	34	adantl	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → 𝑥 ∈ { 0 , 𝑥 } )
36		c0ex	⊢ 0 ∈ V
37	36	prid1	⊢ 0 ∈ { 0 , 𝑥 }
38	35 37	jctil	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → ( 0 ∈ { 0 , 𝑥 } ∧ 𝑥 ∈ { 0 , 𝑥 } ) )
39	9 33 38	rspcedvdw	⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ) → ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) ( 0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒 ) )
40	39	rabeqcda	⊢ ( 𝑁 ∈ ℕ0 → { 𝑥 ∈ ( 𝑉 ∖ { 0 } ) ∣ ∃ 𝑒 ∈ ( Edg ‘ 𝐺 ) ( 0 ∈ 𝑒 ∧ 𝑥 ∈ 𝑒 ) } = ( 𝑉 ∖ { 0 } ) )
41	6 40	eqtrd	⊢ ( 𝑁 ∈ ℕ0 → ( 𝐺 NeighbVtx 0 ) = ( 𝑉 ∖ { 0 } ) )