Description: The base set of a restriction is a subset of the base set of the original structure. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 30-Apr-2015) (Proof shortened by SN, 25-Feb-2025)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ressbas.r | |- R = ( W |`s A ) |
|
| ressbas.b | |- B = ( Base ` W ) |
||
| Assertion | ressbasss | |- ( Base ` R ) C_ B |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ressbas.r | |- R = ( W |`s A ) |
|
| 2 | ressbas.b | |- B = ( Base ` W ) |
|
| 3 | 1 2 | ressbasssg | |- ( Base ` R ) C_ ( A i^i B ) |
| 4 | inss2 | |- ( A i^i B ) C_ B |
|
| 5 | 3 4 | sstri | |- ( Base ` R ) C_ B |