Theorem brric

Description: The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019)

	Ref	Expression
Assertion	brric	⊢ ( 𝑅 ≃𝑟 𝑆 ↔ ( 𝑅 RingIso 𝑆 ) ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		df-ric	⊢ ≃𝑟 = ( ◡ RingIso “ ( V ∖ 1o ) )
2		rimfn	⊢ RingIso Fn ( V × V )
3	1 2	brwitnlem	⊢ ( 𝑅 ≃𝑟 𝑆 ↔ ( 𝑅 RingIso 𝑆 ) ≠ ∅ )