Composition of Mapping with Inclusion is Restriction

Theorem
Let $S$ and $T$ be sets.

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

Let $A \subseteq S$ be a subset of the domain of $S$.

Let $i_A: A \to S$ be the inclusion mapping from $A$ to $S$.

Then:
 * $f \circ i_A = f \restriction_A$

where $f \restriction_A$ denotes the restriction of $f$ to $A$.

Equality of Graph
Let $x \in A$.

All three criteria are seen to be fulfilled.

The result follows from Equality of Mappings.