Inverse Image Mapping Induced by Projection

Theorem
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets.

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

For each $j \in I$, let $\pr_j: S \to S_j$ denote the $j$-th projection.

For each $j \in I$ let $\pr_j^\gets: \powerset {S_i} \to \powerset S$ denote the inverse image mapping induced by $\pr_j$

Then for all $j \in I$, $\pr_j^\gets$ is the mapping defined by:
 * $\forall T \subseteq S_i: \map {\pr_j^\gets} T = \displaystyle \prod_{i \mathop \in I} T_i$

where:
 * $T_i = \begin {cases} T & i = j \\ S_i & i \ne j \end {cases}$

Proof
Let $j \in I$.

Let $T \subseteq S_j$.

Let $\family {T_i}_{i \mathop \in I}$ be the family of sets defined by:
 * $T_i = \begin {cases} T & : i = j \\ S_i & : i \ne j \end {cases}$

Then: