Theorem opiedgval

Description: The set of indexed edges of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 21-Sep-2020)

	Ref	Expression
Assertion	opiedgval	$$⊢G\in \left(\mathrm{V}\times \mathrm{V}\right)\to \mathrm{iEdg}\left(G\right)=2^{nd}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		iedgval	$$⊢\mathrm{iEdg}\left(G\right)=if\left(G\in \left(\mathrm{V}\times \mathrm{V}\right),2^{nd}\left(G\right),\mathrm{ef}\left(G\right)\right)$$
2		iftrue	$$⊢G\in \left(\mathrm{V}\times \mathrm{V}\right)\to if\left(G\in \left(\mathrm{V}\times \mathrm{V}\right),2^{nd}\left(G\right),\mathrm{ef}\left(G\right)\right)=2^{nd}\left(G\right)$$
3	1 2	syl5eq	$$⊢G\in \left(\mathrm{V}\times \mathrm{V}\right)\to \mathrm{iEdg}\left(G\right)=2^{nd}\left(G\right)$$