Theorem gicref

Description: Isomorphism is reflexive. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Assertion	gicref	⊢ ( 𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2	1	idghm	⊢ ( 𝑅 ∈ Grp → ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) )
3		cnvresid	⊢ ◡ ( I ↾ ( Base ‘ 𝑅 ) ) = ( I ↾ ( Base ‘ 𝑅 ) )
4	3 2	eqeltrid	⊢ ( 𝑅 ∈ Grp → ◡ ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) )
5		isgim2	⊢ ( ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpIso 𝑅 ) ↔ ( ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) ∧ ◡ ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpHom 𝑅 ) ) )
6	2 4 5	sylanbrc	⊢ ( 𝑅 ∈ Grp → ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpIso 𝑅 ) )
7		brgici	⊢ ( ( I ↾ ( Base ‘ 𝑅 ) ) ∈ ( 𝑅 GrpIso 𝑅 ) → 𝑅 ≃𝑔 𝑅 )
8	6 7	syl	⊢ ( 𝑅 ∈ Grp → 𝑅 ≃𝑔 𝑅 )