Definition:Mapping Preserves Supremum/Subset

Definition
Let $F$ be a subset of $S_1$.

$f$ preserves supremum of $F$
 * $F$ admits a supremum in $\struct {S_1, \preceq_1}$ implies:
 * $\map {f^\to} F$ admits a supremum in $\struct {S_2, \preceq_2}$ and $\map \sup {\map {f^\to} F} = \map f {\sup F}$