Theorem grlicrcl

Description: Reverse closure of the "is locally isomorphic to" relation for graphs. (Contributed by AV, 9-Jun-2025)

	Ref	Expression
Assertion	grlicrcl	⊢ ( 𝐺 ≃𝑙𝑔𝑟 𝑆 → ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) )

Proof

Step	Hyp	Ref	Expression
1		brgrlic	⊢ ( 𝐺 ≃𝑙𝑔𝑟 𝑆 ↔ ( 𝐺 GraphLocIso 𝑆 ) ≠ ∅ )
2		grlimdmrel	⊢ Rel dom GraphLocIso
3	2	ovprc	⊢ ( ¬ ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) → ( 𝐺 GraphLocIso 𝑆 ) = ∅ )
4	3	necon1ai	⊢ ( ( 𝐺 GraphLocIso 𝑆 ) ≠ ∅ → ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) )
5	1 4	sylbi	⊢ ( 𝐺 ≃𝑙𝑔𝑟 𝑆 → ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) )