# Preimage of Subset of Cartesian Product under Injection from Factor

## Theorem

Let $S$ and $T$ be sets such that $T \ne \O$.

Let $S \times T$ denote their cartesian product.

Let $t \in T$ be given.

Let $j_t \subseteq S \times \paren {S \times T}$ be the injection from $S$ to $S \times T$ defined as:

$\forall s \in \S: \map {j_t} s = \tuple {s, t}$

Let $W \subseteq S \times T$.

Let $V = {j_t}^{-1} \sqbrk W$ denote the preimage of $W$ under $j_t$.

Then:

$V = \set {s: \tuple {s, t} \in W}$

## Proof

 $\displaystyle V$ $=$ $\displaystyle {j_t}^{-1} \sqbrk W$ $\displaystyle$ $=$ $\displaystyle \set {s: \map {j_t} s \in W}$ $\displaystyle$ $=$ $\displaystyle \set {s: \tuple {s, t} \in W}$

$\blacksquare$