There's lots of useful notations that can be used reliably in set notation, such as $\forall_{a\in S} P(a)$, stating that for all $a$ in the set $S$, $P(a)$ is true. For classes it is not so clear. It doesn't seem like one can use "for all" in this way. Instead, I typically see long winded phrasings in the form of "There is a mapping between an element $a$ in class $C$ and a corresponding true predicate $P(a)$.
When writing my notes, I'd like to use a more terse notation. I could use something like $\forall_{a\in C} P(a)$ as an abbreviation, but I'd like to use something more recognizable. Historically there's a good reason why we choose the notations we do, and its not always obvious to a neophyte.
What is the shortest well-established notation for this idea of mapping from any/all elements of a class into other objects? (If $\forall$ is used, great! If not, I'd love to know what is actually used)