Definition:Class Mapping

Definition
Let $A$ and $B$ be classes.

Let $\phi$ be a propositional function such that:
 * $\forall x \in A: \exists ! y \in B: \phi \left({x, y}\right)$

Then the class function from $A$ to $B$, denoted $f_\phi: A \to B$ is defined as:
 * $f_\phi = \left\{{\left({x, y}\right): x \in A \land y \in B \land \phi \left({x, y}\right)}\right\}$

The $\phi$ subscript is optional.

Usually, $\left({x, y}\right) \in f$ is instead written as:
 * $f \left({x}\right) = y$

This is the preferred notation.

Also see

 * Definition:Class Injection
 * Definition:Class Surjection
 * Definition:Class Bijection