# 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 $A$.

## Proof

### Equality of Domains

 $\ds \operatorname{Dom} \left({f \circ i_A}\right)$ $=$ $\ds \operatorname{Dom} \left({i_A}\right)$ Domain of Composite Relation $\ds$ $=$ $\ds A$ by definition of $i_A$ $\ds$ $=$ $\ds \operatorname{Dom} \left({f \restriction_A}\right)$ by definition of restriction of mapping

$\Box$

### Equality of Codomains

 $\ds \operatorname{Cdm} \left({f \circ i_A}\right)$ $=$ $\ds \operatorname{Cdm} \left({f}\right)$ Codomain of Composite Relation $\ds$ $=$ $\ds T$ by definition of $f$ $\ds$ $=$ $\ds \operatorname{Cdm} \left({f \restriction_A}\right)$ by definition of restriction of mapping

$\Box$

### Equality of Graph

Let $x \in A$.

 $\ds \left({f \circ i_A}\right) \left({x}\right)$ $=$ $\ds f \left({i_A \left({x}\right)}\right)$ by definition of composition of mappings $\ds$ $=$ $\ds f \left({x}\right)$ by definition of $i_A$ $\ds$ $=$ $\ds f \restriction_A \left({x}\right)$ by definition of restriction of mapping

$\Box$

All three criteria are seen to be fulfilled.

The result follows from Equality of Mappings.

$\blacksquare$