Equivalence of Definitions of Bijection/Definition 3 iff Definition 4

Necessary Condition
Let $f^{-1}: T \to S$ be a mapping.

Then by definition:
 * $\forall y \in T: \exists x \in S: \tuple {y, x} \in f^{-1}$

Thus for all $y \in T$ there exists at least one $x \in S$ such that $\tuple {y, x} \in f^{-1}$.

Also by definition of mapping:


 * $\tuple {x_1, y} \in f^{-1} \land \tuple {x_2, y} \in f^{-1} \implies x_1 = x_2$

Thus for all $y \in T$ there exists at most one $x \in S$ such that $\tuple {y, x} \in f^{-1}$.

Hence it has been demonstrated that $y \in T$ there exists a unique $x \in S$ such that $\tuple {y, x} \in f^{-1}$.

Sufficient Condition
Let $f$ be such that for all $y \in T$ there exists a unique $x \in S$ such that $\tuple {y, x} \in f^{-1}$.

Then by definition $f^{-1}$ is a mapping.