# Composition of Mapping with Inclusion is Restriction

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## Contents

## 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

\(\displaystyle \operatorname{Dom} \left({f \circ i_A}\right)\) | \(=\) | \(\displaystyle \operatorname{Dom} \left({i_A}\right)\) | Domain of Composite Relation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle A\) | by definition of $i_A$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \operatorname{Dom} \left({f \restriction_A}\right)\) | by definition of restriction of mapping |

$\Box$

### Equality of Codomains

\(\displaystyle \operatorname{Cdm} \left({f \circ i_A}\right)\) | \(=\) | \(\displaystyle \operatorname{Cdm} \left({f}\right)\) | Codomain of Composite Relation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle T\) | by definition of $f$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \operatorname{Cdm} \left({f \restriction_A}\right)\) | by definition of restriction of mapping |

$\Box$

### Equality of Graph

Let $x \in A$.

\(\displaystyle \left({f \circ i_A}\right) \left({x}\right)\) | \(=\) | \(\displaystyle f \left({i_A \left({x}\right)}\right)\) | by definition of composition of mappings | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle f \left({x}\right)\) | by definition of $i_A$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle f \restriction_A \left({x}\right)\) | by definition of restriction of mapping |

$\Box$

All three criteria are seen to be fulfilled.

The result follows from Equality of Mappings.

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
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