Subspace of Product Space is Homeomorphic to Factor Space/Product with Singleton/Lemma

Theorem
Let $T_1$ and $T_2$ be non-empty sets.

Let $b \in T_2$.

Let $T_1 \times T_2$ be the Cartesian product of $T_1$ and $T_2$.

Let $f: T_1 \to T_1 \times \set{b}$ be the mapping defined by:
 * $\map f x = \tuple {x, b}$

Then:
 * $f$ is a bijection.

f is an injection
Let $x, y \in T_1$.

Then $f$ is an injection by definition.

f is a surjection
Let $t \in T_1 \times \set {b}$

Then:
 * $\exists x \in T_1: t = \tuple{x, b}$

So:
 * $\map f x = \tuple{x, b} = t$

Then $f$ is a surjection by definition.

It follows that f is a bijection by definition.