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 = ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩
Assertion	graop	⊢ ( ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐻 ) ∧ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		graop.h	⊢ 𝐻 = ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩
2	1	fveq2i	⊢ ( Vtx ‘ 𝐻 ) = ( Vtx ‘ ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ )
3		fvex	⊢ ( Vtx ‘ 𝐺 ) ∈ V
4		fvex	⊢ ( iEdg ‘ 𝐺 ) ∈ V
5	3 4	opvtxfvi	⊢ ( Vtx ‘ ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ ) = ( Vtx ‘ 𝐺 )
6	2 5	eqtr2i	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐻 )
7	1	fveq2i	⊢ ( iEdg ‘ 𝐻 ) = ( iEdg ‘ ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ )
8	3 4	opiedgfvi	⊢ ( iEdg ‘ ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ ) = ( iEdg ‘ 𝐺 )
9	7 8	eqtr2i	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐻 )
10	6 9	pm3.2i	⊢ ( ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐻 ) ∧ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐻 ) )