Theorem submgmid

Description: Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020)

	Ref	Expression
Hypothesis	submgmss.b	|- B = ( Base ` M )
Assertion	submgmid	|- ( M e. Mgm -> B e. ( SubMgm ` M ) )

Proof

Step	Hyp	Ref	Expression
1		submgmss.b	|- B = ( Base ` M )
2		ssidd	|- ( M e. Mgm -> B C_ B )
3	1	ressid	|- ( M e. Mgm -> ( M |`s B ) = M )
4		id	|- ( M e. Mgm -> M e. Mgm )
5	3 4	eqeltrd	|- ( M e. Mgm -> ( M |`s B ) e. Mgm )
6		eqid	|- ( M |`s B ) = ( M |`s B )
7	1 6	issubmgm2	|- ( M e. Mgm -> ( B e. ( SubMgm ` M ) <-> ( B C_ B /\ ( M |`s B ) e. Mgm ) ) )
8	2 5 7	mpbir2and	|- ( M e. Mgm -> B e. ( SubMgm ` M ) )