Functional predicate

The idea of the functional predicate allows mathematicians to extend the concept of function beyond specific sets.

A functional predicate $F(x,y)$ is a predicate in two variables (in this case, $x$ and $y$) such that $F(x,y)$ and $F(x,z)$ together imply $y=z$. If $F(x,y)$ holds, then we may write $F(x) = y$.

Note that this permits us to speak of "functions" which act on all sets. In set theory, the relation $f(x) = \{x\}$, for example, cannot yield a function unless it is confined to a specific domain and range. However, we may speak of the functional predicate $F(x,y) : = (y = \{x\})$, which may be applied to any set. This permits us to speak of general functions on sets within the context of classical set theory.

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