Composition of Mapping and Inclusion is Restriction of Mapping

Theorem
Let $S, T$ be sets.

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

Let $A \subseteq S$.

Then $f \circ i_A = f \restriction A$

where
 * $i_A$ denotes the inclusion mapping of $A$,
 * $f \restriction A$ denotes the restriction of $f$ to $A$.

Proof
By definition of inclusion mapping:
 * $i_A: A \to S$

By definitions of composition of mappings and restriction of mapping:
 * $f \circ i_A: A \to T$ and $f \restriction A: A \to T$

Let $a \in A$.

Thus