Theorem submgmbas

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

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

Proof

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