Restriction of Mapping is its Intersection with Cartesian Product of Subset with Image

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

Let $X \subseteq S$.

Let $f {\restriction_X}$ be the restriction of $f$ to $X$.

Then:
 * $f {\restriction_X} = f \cap \paren {X \times \Img f}$

where:
 * $\Img f$ denotes the image of $f$, defined as:
 * $\Img f = \set {t \in T: \exists s \in S: t = \map f s}$
 * $X \times \Img f$ denotes the cartesian product of $X$ with $\Img f$.

Proof
We have:

Then:

Thus we have: