Number of Bijective Restrictions

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

Let $B$ be the set of all bijective restrictions of $f$.

Then the cardinality of $B$ is:


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

where $S / \mathcal{R}_f$ denotes the quotient set of the induced equivalence of $f$ indexed by $I$.