Description: Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020)
Ref | Expression | ||
---|---|---|---|
Hypothesis | submgmss.b | |- B = ( Base ` M ) |
|
Assertion | submgmss | |- ( S e. ( SubMgm ` M ) -> S C_ B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submgmss.b | |- B = ( Base ` M ) |
|
2 | submgmrcl | |- ( S e. ( SubMgm ` M ) -> M e. Mgm ) |
|
3 | eqid | |- ( M |`s S ) = ( M |`s S ) |
|
4 | 1 3 | issubmgm2 | |- ( M e. Mgm -> ( S e. ( SubMgm ` M ) <-> ( S C_ B /\ ( M |`s S ) e. Mgm ) ) ) |
5 | 2 4 | syl | |- ( S e. ( SubMgm ` M ) -> ( S e. ( SubMgm ` M ) <-> ( S C_ B /\ ( M |`s S ) e. Mgm ) ) ) |
6 | 5 | ibi | |- ( S e. ( SubMgm ` M ) -> ( S C_ B /\ ( M |`s S ) e. Mgm ) ) |
7 | 6 | simpld | |- ( S e. ( SubMgm ` M ) -> S C_ B ) |