Theorem subgrcl

Description: Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014)

	Ref	Expression
Assertion	subgrcl	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
2	1	issubg	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ↔ ( 𝐺 ∈ Grp ∧ 𝑆 ⊆ ( Base ‘ 𝐺 ) ∧ ( 𝐺 ↾s 𝑆 ) ∈ Grp ) )
3	2	simp1bi	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp )