Cardinality of Cartesian Product of Finite Sets/Corollary/Proof 2

Corollary to Cardinality of Cartesian Product
Let $S \times T$ be the cartesian product of two sets $S$ and $T$ which are both finite.

Then:
 * $\left|{S \times T}\right| = \left|{T \times S}\right|$

where $\left|{S \times T}\right|$ denotes the cardinality of $S \times T$.

Proof
Let $f: S \times T \to T \times S$ be the mapping defined as:
 * $\forall \left({s, t}\right) \in S \times T: f \left({s, t}\right) = \left({t, s}\right)$

which is shown to be bijective as follows:

showing $f$ is an injection.

Let $\left({t, s}\right) \in T \times S$.

Then:
 * $\exists \left({s, t}\right) \in S \times T: f \left({s, t}\right) = \left({t, s}\right)$

showing that $f$ is a surjection.

So we have demonstrated that there exists a bijection from $S \times T$ to $T \times S$.

The result follows by definition of set equivalence.