Number of Injective Restrictions

Theorem
Let $f:S \to T$ be a mapping.

Let $P$ be the set of all injective restrictions of $f$.

Then the cardinality of $P$ is:


 * $\displaystyle \left \vert{\prod_{i \in I} \prod_{j \in J_i} \left ({\left({\mathcal{P}\left({S / \mathcal{R}_f}\right)}\right)_i}\right)_j} \right \vert$

where $\mathcal{P}\left({S / \mathcal{R}_f}\right)$ denotes the power set of the quotient set of the induced equivalence of $f$ whose elements are indexed by $I$ with each element being further indexed by an index set $J_i$ where $i \in I$.