Projection is Surjection/General Version

Theorem
For all non-empty sets $S_1, S_2, \ldots, S_j, \ldots, S_n$, the $j$th projection $\operatorname{pr}_j$ on $\displaystyle \prod_{i=1}^n S_i$ is a surjection.

Proof
Consider the $j$th projection.

As long as none of $S_1, S_2, \ldots, S_n$ is the empty set, then:


 * $\displaystyle \forall x \in S_j: \exists \left({s_1, s_2, \ldots, s_{j-1}, x, s_{j+1}, \ldots, s_n}\right) \in \prod_{k=1}^n S_k: \operatorname{pr}_j \left({\left({s_1, s_2, \ldots, s_{j-1}, x, s_{j+1}, \ldots, s_n}\right)}\right) = x$

Hence the result.