Mapping is Injection and Surjection iff Inverse is Mapping

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Then:
 * $f: S \to T$ can be defined as a bijection in the sense that:
 * $(1): \quad f$ is an injection
 * $(2): \quad f$ is a surjection.

iff:


 * the inverse $f^{-1}$ of $f$ is such that:
 * for each $y \in T$, the preimage $f^{-1} \left({\left\{{y}\right\}}\right)$ has exactly one element.


 * That is, such that $f^{-1} \subseteq T \times S$ is itself a mapping.

Proof
Let $f: S \to T$ be a mapping.

Recall the definition of the inverse of $f$:

$f^{-1} \subseteq T \times S$ is the relation defined as:
 * $f^{-1} = \left\{{\left({t, s}\right): t = f \left({s}\right)}\right\}$

Necessary Condition
Let $f: S \to T$ be a bijection in the sense that:
 * $(1): \quad f$ is an injection
 * $(2): \quad f$ is a surjection.

By Inverse of Injection is Functional Relation‎, $f^{-1}$ is functional.

That is:
 * $\forall y_1, y_2 \in T: f^{-1} \left({y_1}\right) \ne f^{-1} \left({y_2}\right) \implies y_1 = y_2$

Hence the preimage $f^{-1} \left({\left\{{y}\right\}}\right)$ has at most one element.

By Surjection iff Image equals Codomain:
 * $\operatorname{Im} \left({f}\right) = T$

That is:
 * $\forall y \in T: \exists x \in S: f^{-1} \left({y}\right) = x$

Hence the preimage $f^{-1} \left({\left\{{y}\right\}}\right)$ has at least one element.

Thus the preimage $f^{-1} \left({\left\{{y}\right\}}\right)$ has exactly one element.

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

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

It is necessary to show that $f$ is both an injection and a surjection.

Let $f \left({x_a}\right) = y$ and $f \left({x_b}\right) = y$.

Then:

Thus, by definition, $f$ is an injection.

Aiming for a contradiction, suppose that $f$ is not a surjection.

That is:
 * $\exists y \in T: \neg \exists x \in S: \left({x, y}\right) \in f$

By definition of inverse of mapping:
 * $\exists y \in T: \neg \exists x \in S: \left({y, x}\right) \in f^{-1}$

which would mean that $f^{-1}$ is not a mapping.

From this contradiction it follows that $f$ is a surjection.

So, being both an injection and a surjection, it follows by definition that $f$ is a bijection.