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 e. NN0 -> 0 e. V )

Proof

Step	Hyp	Ref	Expression
1		stgrvtx0.g	|- G = ( StarGr ` N )
2		stgrvtx0.v	|- V = ( Vtx ` G )
3		stgrvtx	|- ( N e. NN0 -> ( 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 e. NN0 -> 0 e. ( 0 ... N ) )
8		eleq2	|- ( V = ( 0 ... N ) -> ( 0 e. V <-> 0 e. ( 0 ... N ) ) )
9	7 8	syl5ibrcom	|- ( N e. NN0 -> ( V = ( 0 ... N ) -> 0 e. V ) )
10	6 9	biimtrrid	|- ( N e. NN0 -> ( ( Vtx ` ( StarGr ` N ) ) = ( 0 ... N ) -> 0 e. V ) )
11	3 10	mpd	|- ( N e. NN0 -> 0 e. V )