Equivalence of Definitions of Bijection/Definition 1 iff Definition 4

Proof
Let $f: S \to T$ be a bijection by definition 1.

Then by definition:
 * $f$ is an injection
 * $f$ is a surjection

By definition of injection:
 * every element of $T$ is the image of at most $1$ element of $S$.

By definition of surjection:
 * every element of $T$ is the image of at least $1$ element of $S$.

So:
 * for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.

Thus $f$ is a bijection by definition 4.

Let $f: S \to T$ be a bijection by definition 4.

Then by definition:
 * for each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.

But:
 * every element of $T$ is the image of at most $1$ element of $S$

defines an injection

and:
 * every element of $T$ is the image of at least $1$ element of $S$

defines a surjection.

From Injection iff Left Inverse, $f$ is an injection $f$ has a left inverse mapping.

From Surjection iff Right Inverse, $f$ is a surjection $f$ has a right inverse mapping.

Putting these together, it follows that:
 * $f$ is an injection
 * $f$ is a surjection

Thus $f$ is a bijection by definition 1.