Theorem brgici

Description: Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015)

	Ref	Expression
Assertion	brgici	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝑅 ≃𝑔 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		ne0i	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) → ( 𝑅 GrpIso 𝑆 ) ≠ ∅ )
2		brgic	⊢ ( 𝑅 ≃𝑔 𝑆 ↔ ( 𝑅 GrpIso 𝑆 ) ≠ ∅ )
3	1 2	sylibr	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) → 𝑅 ≃𝑔 𝑆 )