Theorem submgmss

Description: Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020)

	Ref	Expression
Hypothesis	submgmss.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
Assertion	submgmss	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑆 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		submgmss.b	⊢ 𝐵 = ( Base ‘ 𝑀 )
2		submgmrcl	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑀 ∈ Mgm )
3		eqid	⊢ ( 𝑀 ↾s 𝑆 ) = ( 𝑀 ↾s 𝑆 )
4	1 3	issubmgm2	⊢ ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mgm ) ) )
5	2 4	syl	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ 𝐵 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mgm ) ) )
6	5	ibi	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ⊆ 𝐵 ∧ ( 𝑀 ↾s 𝑆 ) ∈ Mgm ) )
7	6	simpld	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑆 ⊆ 𝐵 )