Theorem submbas

Description: The base set of a submonoid. (Contributed by Stefan O'Rear, 15-Jun-2015)

	Ref	Expression
Hypothesis	submmnd.h	|- H = ( M |`s S )
Assertion	submbas	|- ( S e. ( SubMnd ` M ) -> S = ( Base ` H ) )

Proof

Step	Hyp	Ref	Expression
1		submmnd.h	|- H = ( M |`s S )
2		eqid	|- ( Base ` M ) = ( Base ` M )
3	2	submss	|- ( S e. ( SubMnd ` M ) -> S C_ ( Base ` M ) )
4	1 2	ressbas2	|- ( S C_ ( Base ` M ) -> S = ( Base ` H ) )
5	3 4	syl	|- ( S e. ( SubMnd ` M ) -> S = ( Base ` H ) )