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:


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

where:
 * $\mathcal P$ denotes power set
 * $S / \mathcal R_f$ denotes quotient set of the induced equivalence of $f$.