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 $j_t: S \to S \times T$ 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:
 * $V = \set {s \in S : \tuple {s, t} \in W}$

Proof
That $j_t$ is actually an injection is demonstrated in Mapping from Set to Ordinate of Cartesian Product is Injection.

Then: