With Set Builder Notation, one usually create sets of individuals based on some conditions, eg. $S=\{x|P(x)\}$ as the set of all individuals that satisfy $P(x)$.
I was wondering if we could use Set Builder Notation to create kind of the opposite: sets of properties based on an individual. For instance, if we consider $P_1a, \cdots, P_na$ properties that $a$ satisfies, could we construct the set $P=\{Px|Pa\}$ ?
Moreover, could we create a set that include all properties of any arity that $a$ satisfy? Maybe with something like $P'=\{Px|a \in x\}$?