Composition of Mapping with Inclusion is Restriction

From ProofWiki
Jump to navigation Jump to search

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