Composition of Mapping with Inclusion is Restriction

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July 19, 2022:  It has been suggested that this page or section be merged into Composition of Mapping and Inclusion is Restriction of Mapping.
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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.

$\blacksquare$


Sources

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