Theorem gricrcl

Description: Reverse closure of the "is isomorphic to" relation for graphs. (Contributed by AV, 12-Jun-2025)

	Ref	Expression
Assertion	gricrcl	⊢ ( 𝐺 ≃𝑔𝑟 𝑆 → ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) )

Proof

Step	Hyp	Ref	Expression
1		brgric	⊢ ( 𝐺 ≃𝑔𝑟 𝑆 ↔ ( 𝐺 GraphIso 𝑆 ) ≠ ∅ )
2		grimdmrel	⊢ Rel dom GraphIso
3	2	ovprc	⊢ ( ¬ ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) → ( 𝐺 GraphIso 𝑆 ) = ∅ )
4	3	necon1ai	⊢ ( ( 𝐺 GraphIso 𝑆 ) ≠ ∅ → ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) )
5	1 4	sylbi	⊢ ( 𝐺 ≃𝑔𝑟 𝑆 → ( 𝐺 ∈ V ∧ 𝑆 ∈ V ) )