Theorem graop

Description: Any representation of a graph G (especially as extensible structure G = { <. ( Basendx ) , V >. , <. ( .efndx ) , E >. } ) is convertible in a representation of the graph as ordered pair. (Contributed by AV, 7-Oct-2020)

	Ref	Expression
Hypothesis	graop.h	$$⊢H=⟨\mathrm{Vtx}\left(G\right),\mathrm{iEdg}\left(G\right)⟩$$
Assertion	graop	$$⊢\left(\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(H\right)\wedge \mathrm{iEdg}\left(G\right)=\mathrm{iEdg}\left(H\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		graop.h	$$⊢H=⟨\mathrm{Vtx}\left(G\right),\mathrm{iEdg}\left(G\right)⟩$$
2	1	fveq2i	$$⊢\mathrm{Vtx}\left(H\right)=\mathrm{Vtx}\left(⟨\mathrm{Vtx}\left(G\right),\mathrm{iEdg}\left(G\right)⟩\right)$$
3		fvex	$$⊢\mathrm{Vtx}\left(G\right)\in \mathrm{V}$$
4		fvex	$$⊢\mathrm{iEdg}\left(G\right)\in \mathrm{V}$$
5	3 4	opvtxfvi	$$⊢\mathrm{Vtx}\left(⟨\mathrm{Vtx}\left(G\right),\mathrm{iEdg}\left(G\right)⟩\right)=\mathrm{Vtx}\left(G\right)$$
6	2 5	eqtr2i	$$⊢\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(H\right)$$
7	1	fveq2i	$$⊢\mathrm{iEdg}\left(H\right)=\mathrm{iEdg}\left(⟨\mathrm{Vtx}\left(G\right),\mathrm{iEdg}\left(G\right)⟩\right)$$
8	3 4	opiedgfvi	$$⊢\mathrm{iEdg}\left(⟨\mathrm{Vtx}\left(G\right),\mathrm{iEdg}\left(G\right)⟩\right)=\mathrm{iEdg}\left(G\right)$$
9	7 8	eqtr2i	$$⊢\mathrm{iEdg}\left(G\right)=\mathrm{iEdg}\left(H\right)$$
10	6 9	pm3.2i	$$⊢\left(\mathrm{Vtx}\left(G\right)=\mathrm{Vtx}\left(H\right)\wedge \mathrm{iEdg}\left(G\right)=\mathrm{iEdg}\left(H\right)\right)$$