Theorem edgstruct

Description: The edges of a graph represented as an extensible structure with vertices as base set and indexed edges. (Contributed by AV, 13-Oct-2020)

	Ref	Expression
Hypothesis	edgstruct.s	$$⊢G=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},V⟩,⟨\mathrm{ef}\left(\mathrm{ndx}\right),E⟩\right\}$$
Assertion	edgstruct	$$⊢\left(V\in W\wedge E\in X\right)\to \mathrm{Edg}\left(G\right)=\mathrm{ran}E$$

Proof

Step	Hyp	Ref	Expression
1		edgstruct.s	$$⊢G=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},V⟩,⟨\mathrm{ef}\left(\mathrm{ndx}\right),E⟩\right\}$$
2		edgval	$$⊢\mathrm{Edg}\left(G\right)=\mathrm{ran}\mathrm{iEdg}\left(G\right)$$
3	1	struct2griedg	$$⊢\left(V\in W\wedge E\in X\right)\to \mathrm{iEdg}\left(G\right)=E$$
4	3	rneqd	$$⊢\left(V\in W\wedge E\in X\right)\to \mathrm{ran}\mathrm{iEdg}\left(G\right)=\mathrm{ran}E$$
5	2 4	syl5eq	$$⊢\left(V\in W\wedge E\in X\right)\to \mathrm{Edg}\left(G\right)=\mathrm{ran}E$$