Projection is Surjection

Theorem
Let $S$ and $T$ be non-empty sets.

Let $S \times T$ be the Cartesian product of $S$ and $T$.

Let $\pr_1: S \times T \to T$ and $\pr_2: S \times T \to T$ be the first projection and second projection respectively on $S \times T$.

Then $\pr_1$ and $\pr_2$ are both surjections.

Proof
Let $S$ and $T$ be sets such that neither is empty.

Let $\pr_1: S \times T \to S$ be the first projection on $S \times T$.

Then by definition of first projection:
 * $\displaystyle \forall x \in S: \exists \tuple {x, t} \in S \times T: \map {\pr_1} {x, t} = x$

Similarly, let $\pr_2: S \times T \to T$ be the second projection on $S \times T$.

Then by definition of second projection:
 * $\displaystyle \forall x \in T: \exists \tuple {s, x} \in S \times T: \map {\pr_2} {s, x} = x$

Hence the result.