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:

$\ds \card {\prod_{i \mathop \in I} \family {S / \RR_f}_i}$

where $S / \RR_f$ denotes the quotient set of the induced equivalence of $f$ indexed by $I$.