Theorem submbas

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

	Ref	Expression
Hypothesis	submmnd.h	$$⊢H=M{\upharpoonright }_{𝑠}S$$
Assertion	submbas	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to S={\mathrm{Base}}_H$$

Proof

Step	Hyp	Ref	Expression
1		submmnd.h	$$⊢H=M{\upharpoonright }_{𝑠}S$$
2		eqid	$$⊢{\mathrm{Base}}_M={\mathrm{Base}}_M$$
3	2	submss	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to S\subseteq {\mathrm{Base}}_M$$
4	1 2	ressbas2	$$⊢S\subseteq {\mathrm{Base}}_M\to S={\mathrm{Base}}_H$$
5	3 4	syl	$$⊢S\in \mathrm{SubMnd}\left(M\right)\to S={\mathrm{Base}}_H$$