Theorem struct2grvtx

Description: The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 23-Sep-2020)

	Ref	Expression
Hypothesis	struct2grvtx.g	$$⊢G=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},V⟩,⟨\mathrm{ef}\left(\mathrm{ndx}\right),E⟩\right\}$$
Assertion	struct2grvtx	$$⊢\left(V\in X\wedge E\in Y\right)\to \mathrm{Vtx}\left(G\right)=V$$

Proof

Step	Hyp	Ref	Expression
1		struct2grvtx.g	$$⊢G=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},V⟩,⟨\mathrm{ef}\left(\mathrm{ndx}\right),E⟩\right\}$$
2		edgfndxnn	$$⊢\mathrm{ef}\left(\mathrm{ndx}\right)\in \mathbb{N}$$
3		basendxltedgfndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}<\mathrm{ef}\left(\mathrm{ndx}\right)$$
4	2 3 1	structvtxval	$$⊢\left(V\in X\wedge E\in Y\right)\to \mathrm{Vtx}\left(G\right)=V$$