First Projection on Ordered Pair of Sets

Theorem
Let $a$ and $b$ be sets.

Let $w = \tuple {a, b}$ denote the ordered pair of $a$ and $b$.

Let $\map {\pr_1} w$ denote the first projection on $w$.

Then:
 * $\displaystyle \map {\pr_1} w = \bigcup \bigcap w$

where $\displaystyle \bigcup$ and $\displaystyle \bigcap$ denote union and intersection respectively.

Proof
We have by definition of first projection that:
 * $\map {\pr_1} w = \map {\pr_1} {a, b} = a$

Then: