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 $\pr_1: A \times B \to A$ be the first projection of $A \times B$.

Let $a \in A$.

Then:

$\pr_1^{-1} \sqbrk {\set a} = \set {\tuple {a, b}: b \in B}$

that is:

$\pr_1^{-1} \sqbrk {\set a} = \set a \times B$

Proof

Directly apparent from the definition of cartesian product.

$\blacksquare$