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?
Yes, each set $x$ corresponds to a unary predicate $\varphi_x(y) := y\in x$ which is a projection of the membership relation, which is a binary predicate $\varphi_\in(x,y) := x\in y.$
And yes, while sets correspond to predicates, not all predicates correspond to a set. For instance, $\{(x,y): x\in y\}$ is not a set. Collections of sets that do not correspond to a set are called proper classes.
(Note, I've used "set-builder" notation here, which is really "class-builder" notation, to describe classes. In set theories like ZFC where we don't have a formal notion of class, classes just correspond to predicates. For instance, $\{x:\varphi(x)\}$ just corresponds to the predicate $\varphi(x).$ And since we have an ordered pairing function, one doesn't really need to distinguish between predicates of different arities: $\{(x,y): x\in y\}$ can either be thought of as a binary predicate $x\in y$ or as a unary predicate "$z$ is an ordered pair whose first element is $x$ and second element is $y$ and $x\in y.$)
Similarly, many other logical functions and relations are proper classes, like the collection of all sets , the function $F(x,y) = \{x,y\},$ etc. But sometimes, like in the case of $X\times Y =\{(x,y): x\in X\land y\in Y\}$ we can show from the axioms that there is a set whose elements are all sets satisfying the predicate.