In so many different instances we need to be able to construct sets of functions. The first axiom of set theory (at least in the order that I learned) says that $a\in b$ is only a proposition if a and b are both sets. Thus if a relation (and furthermore a function) is an element of a set then it too must be a set by this axiom. This is fine because half of the time we describe relations as subsets of some Cartesian product $X\times Y$. The other half, though, we say relations are predicates of two variables, i.e. $R(x,y)$ is true for certain $x$ and $y$ and false for others. If we combine these descriptions, we get what I call the "set-predicate correspondence axiom" that says if $a$ is an element of $b$ then $b(a)$ is true and if a is not an element of $b$ then $b(a)$ is false. In other words, sets are predicates though not necessarily the other way around. This is the only way I could think of to allow constructions of sets of predicates such as relations, functions, etc.
Has this axiom already been established? Do we at least accept this idea in mathematics? If not, how do we compensate for the discrepancy between two definitions of relations and the desire to construct sets of functions (or other predicates) as elements?