Inclusion Mapping is Restriction of Identity

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Theorem

Let $T$ be a set.

Let $S \subseteq T$ be a subset of $T$.

Let $i_S: S \to T$ be the inclusion mapping on $S$.


Then $i_S$ is the restriction of the identity mapping $I_T: T \to T$ on $T$.


Proof

By definition of inclusion mapping:

$i_S: S \to T: \forall x \in S: \map {i_S} x = x$

By definition of identity mapping:

$I_T: T \to T: \forall x \in T: \map {I_T} x = x$

The result follows by definition of restriction of mapping.

$\blacksquare$


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