Theorem usgrop

Description: A simple graph represented by an ordered pair. (Contributed by AV, 23-Oct-2020) (Proof shortened by AV, 30-Nov-2020)

	Ref	Expression
Assertion	usgrop	⊢ ( 𝐺 ∈ USGraph → ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ ∈ USGraph )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
2		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
3	1 2	usgrfs	⊢ ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } )
4		fvex	⊢ ( Vtx ‘ 𝐺 ) ∈ V
5		fvex	⊢ ( iEdg ‘ 𝐺 ) ∈ V
6	4 5	pm3.2i	⊢ ( ( Vtx ‘ 𝐺 ) ∈ V ∧ ( iEdg ‘ 𝐺 ) ∈ V )
7		isusgrop	⊢ ( ( ( Vtx ‘ 𝐺 ) ∈ V ∧ ( iEdg ‘ 𝐺 ) ∈ V ) → ( ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
8	6 7	mp1i	⊢ ( 𝐺 ∈ USGraph → ( ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) : dom ( iEdg ‘ 𝐺 ) –1-1→ { 𝑥 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ∣ ( ♯ ‘ 𝑥 ) = 2 } ) )
9	3 8	mpbird	⊢ ( 𝐺 ∈ USGraph → ⟨ ( Vtx ‘ 𝐺 ) , ( iEdg ‘ 𝐺 ) ⟩ ∈ USGraph )