Theorem isgim2

Description: A group isomorphism is a homomorphism whose converse is also a homomorphism. Characterization of isomorphisms similar to ishmeo . (Contributed by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	isgim2	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
3	1 2	isgim	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) )
4	1 2	ghmf1o	⊢ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) → ( 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ↔ ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ) )
5	4	pm5.32i	⊢ ( ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ) )
6	3 5	bitri	⊢ ( 𝐹 ∈ ( 𝑅 GrpIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 GrpHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑅 ) ) )