# 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

 $\ds \Dom {f \circ i_A}$ $=$ $\ds \Dom {i_A}$ Domain of Composite Relation $\ds$ $=$ $\ds A$ Definition of $i_A$ $\ds$ $=$ $\ds \Dom {f \restriction_A}$ Definition of Restriction of Mapping

$\Box$

### Equality of Codomains

 $\ds \Cdm {f \circ i_A}$ $=$ $\ds \Cdm f$ Codomain of Composite Relation $\ds$ $=$ $\ds T$ Definition of $f$ $\ds$ $=$ $\ds \Cdm {f \restriction_A}$ Definition of Restriction of Mapping

$\Box$

### Equality of Graph

Let $x \in A$.

 $\ds \map {\paren {f \circ i_A} } x$ $=$ $\ds \map f {\map {i_A} x}$ Definition of Composition of Mappings $\ds$ $=$ $\ds \map f x$ Definition of $i_A$ $\ds$ $=$ $\ds \map {f \restriction_A} x$ Definition of Restriction of Mapping

$\Box$

All three criteria are seen to be fulfilled.

The result follows from Equality of Mappings.

$\blacksquare$