Theorem gimcnv

Description: The converse of a bijective group homomorphism is a bijective group homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015) (Revised by Mario Carneiro, 6-May-2015)

	Ref	Expression
Assertion	gimcnv	⊢ ( 𝐹 ∈ ( 𝑆 GrpIso 𝑇 ) → ◡ 𝐹 ∈ ( 𝑇 GrpIso 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
2		eqid	⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
3	1 2	ghmf	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) )
4		frel	⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) → Rel 𝐹 )
5		dfrel2	⊢ ( Rel 𝐹 ↔ ◡ ◡ 𝐹 = 𝐹 )
6	4 5	sylib	⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) → ◡ ◡ 𝐹 = 𝐹 )
7	3 6	syl	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → ◡ ◡ 𝐹 = 𝐹 )
8		id	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) )
9	7 8	eqeltrd	⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → ◡ ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) )
10	9	anim1ci	⊢ ( ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 GrpHom 𝑆 ) ) → ( ◡ 𝐹 ∈ ( 𝑇 GrpHom 𝑆 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) )
11		isgim2	⊢ ( 𝐹 ∈ ( 𝑆 GrpIso 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 GrpHom 𝑆 ) ) )
12		isgim2	⊢ ( ◡ 𝐹 ∈ ( 𝑇 GrpIso 𝑆 ) ↔ ( ◡ 𝐹 ∈ ( 𝑇 GrpHom 𝑆 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) )
13	10 11 12	3imtr4i	⊢ ( 𝐹 ∈ ( 𝑆 GrpIso 𝑇 ) → ◡ 𝐹 ∈ ( 𝑇 GrpIso 𝑆 ) )