Equivalence of Definitions of Bijection/Definition 1 iff Definition 3

Theorem

The following definitions of the concept of Bijection are equivalent:

Definition 1

A mapping $f: S \to T$ is a bijection if and only if both:

$(1): \quad f$ is an injection

and:

$(2): \quad f$ is a surjection.

Definition 3

A mapping $f: S \to T$ is a bijection if and only if:

the inverse $f^{-1}$ of $f$ is a mapping from $T$ to $S$.

Proof

This is demonstrated in Mapping is Injection and Surjection iff Inverse is Mapping.

$\blacksquare$