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 $\operatorname{pr}_1: S \times T \to T$ and $\operatorname{pr}_2: S \times T \to T$ be the first projection and second projection respectively on $S \times T$.

Then $\operatorname{pr}_1$ and $\operatorname{pr}_2$ are both surjections.

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

Let $\operatorname{pr}_1: S \times T \to S$ be the first projection on $S \times T$.

Then by definition of first projection:
 * $\displaystyle \forall x \in S: \exists \left({x, t}\right) \in S \times T: \operatorname{pr}_1 \left({\left({x, t}\right)}\right) = x$

Similarly, let $\operatorname{pr}_2: S \times T \to T$ be the second projection on $S \times T$.

Then by definition of second projection:
 * $\displaystyle \forall x \in T: \exists \left({s, x}\right) \in S \times T: \operatorname{pr}_2 \left({\left({s, x}\right)}\right) = x$

Hence the result.