# 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

 $\ds \map {\paren {f \circ i_A} } a$ $=$ $\ds \map f {\map {i_A} a}$ Definition of Composition of Mappings $\ds$ $=$ $\ds \map f a$ Definition of Inclusion Mapping $\ds$ $=$ $\ds \map {\paren {f \restriction A} } a$ Definition of Restriction of Mapping

$\blacksquare$