Theorem grimid

Description: The identity relation restricted to the set of vertices of a graph is a graph isomorphism between the graph and itself. (Contributed by AV, 29-Apr-2025) (Prove shortened by AV, 5-May-2025.)

	Ref	Expression
Assertion	grimid	⊢ ( 𝐺 ∈ UHGraph → ( I ↾ ( Vtx ‘ 𝐺 ) ) ∈ ( 𝐺 GraphIso 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph )
2		eqidd	⊢ ( 𝐺 ∈ UHGraph → ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) )
3		eqidd	⊢ ( 𝐺 ∈ UHGraph → ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 ) )
4	1 1 2 3	grimidvtxedg	⊢ ( 𝐺 ∈ UHGraph → ( I ↾ ( Vtx ‘ 𝐺 ) ) ∈ ( 𝐺 GraphIso 𝐺 ) )