Theorem brrici

Description: Prove isomorphic by an explicit isomorphism. (Contributed by SN, 10-Jan-2025)

	Ref	Expression
Assertion	brrici	⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝑅 ≃𝑟 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		ne0i	⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → ( 𝑅 RingIso 𝑆 ) ≠ ∅ )
2		brric	⊢ ( 𝑅 ≃𝑟 𝑆 ↔ ( 𝑅 RingIso 𝑆 ) ≠ ∅ )
3	1 2	sylibr	⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝑅 ≃𝑟 𝑆 )