Theorem submgmbas

Description: The base set of a submagma. (Contributed by AV, 26-Feb-2020)

	Ref	Expression
Hypothesis	submgmmgm.h	⊢ 𝐻 = ( 𝑀 ↾s 𝑆 )
Assertion	submgmbas	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑆 = ( Base ‘ 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		submgmmgm.h	⊢ 𝐻 = ( 𝑀 ↾s 𝑆 )
2		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
3	2	submgmss	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑆 ⊆ ( Base ‘ 𝑀 ) )
4	1 2	ressbas2	⊢ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) → 𝑆 = ( Base ‘ 𝐻 ) )
5	3 4	syl	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑆 = ( Base ‘ 𝐻 ) )