Subspace of Product Space is Homeomorphic to Factor Space/Proof 1/Lemma 2

Theorem
Let $\family {X_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set.

Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the Cartesian product of $\family {X_i}_{i \mathop \in I}$.

Let $z \in X$.

Let $i \in I$.

Let $\pr_i : X \to X_i$ be the $i$th-projection from $X$.

For all for all $j \in I$ let:
 * $Z_j = \begin{cases} X_i & i = j \\

\set{z_j} & j \neq i \end{cases}$

Let $Y_i = \prod_{j \mathop \in I} Z_j$

Let $p_i : Y_i \to X_i$ be the $i$th-projection from $Y_i$.

Then:
 * $\pr_i {\restriction_{Y_i} } = p_i$

Proof
For all $y \in Y_i$:

By equality of mappings, $\pr_i {\restriction_{Y_i} } = p_i$