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Metamath Proof Explorer

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	$⊢ G \in UHGraph \to {I ↾}_{Vtx (G)} \in (G GraphIso G)$

Proof

Step	Hyp	Ref	Expression
1		id	$⊢ G \in UHGraph \to G \in UHGraph$
2		eqidd	$⊢ G \in UHGraph \to Vtx (G) = Vtx (G)$
3		eqidd	$⊢ G \in UHGraph \to iEdg (G) = iEdg (G)$
4	1 1 2 3	grimidvtxedg	$⊢ G \in UHGraph \to {I ↾}_{Vtx (G)} \in (G GraphIso G)$