Theorem funvtxval

Description: The set of vertices of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 22-Sep-2020) (Revised by AV, 7-Jun-2021) (Revised by AV, 12-Nov-2021)

	Ref	Expression
Assertion	funvtxval	$$⊢\left(\mathrm{Fun}\left(G\setminus \left\{\varnothing \right\}\right)\wedge \left\{{\mathrm{Base}}_{\mathrm{ndx}},\mathrm{ef}\left(\mathrm{ndx}\right)\right\}\subseteq \mathrm{dom}G\right)\to \mathrm{Vtx}\left(G\right)={\mathrm{Base}}_G$$

Proof

Step	Hyp	Ref	Expression
1		basendxnedgfndx	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{ef}\left(\mathrm{ndx}\right)$$
2		fvex	$$⊢{\mathrm{Base}}_{\mathrm{ndx}}\in \mathrm{V}$$
3		fvex	$$⊢\mathrm{ef}\left(\mathrm{ndx}\right)\in \mathrm{V}$$
4	2 3	funvtxdm2val	$$⊢\left(\mathrm{Fun}\left(G\setminus \left\{\varnothing \right\}\right)\wedge {\mathrm{Base}}_{\mathrm{ndx}}\ne \mathrm{ef}\left(\mathrm{ndx}\right)\wedge \left\{{\mathrm{Base}}_{\mathrm{ndx}},\mathrm{ef}\left(\mathrm{ndx}\right)\right\}\subseteq \mathrm{dom}G\right)\to \mathrm{Vtx}\left(G\right)={\mathrm{Base}}_G$$
5	1 4	mp3an2	$$⊢\left(\mathrm{Fun}\left(G\setminus \left\{\varnothing \right\}\right)\wedge \left\{{\mathrm{Base}}_{\mathrm{ndx}},\mathrm{ef}\left(\mathrm{ndx}\right)\right\}\subseteq \mathrm{dom}G\right)\to \mathrm{Vtx}\left(G\right)={\mathrm{Base}}_G$$