Preimage of Element under Projection

Theorem
Let $A$ and $B$ be sets.

Let $A \times B$ be the cartesian product of $A$ and $B$.

Let $\operatorname{pr}_1: A \times B \to A$ be the first projection of $A \times B$.

Let $a \in A$.

Then:
 * $\operatorname{pr}_1^{-1} \left[{\left\{{a}\right\}}\right] = \left\{{\left({a, b}\right): b \in B}\right\}$

Proof
Directly apparent from the definition of cartesian product.