Identity Mapping is Left Identity

Theorem
Let $$\mathcal R \subseteq S \times T$$ be a relation on $$S \times T$$.

Then:
 * $$I_T \circ \mathcal R = \mathcal R$$

where $$I_T$$ is the identity mapping on $$T$$, and $$\circ$$ signifies composition of relations.

Corollary
Let $$f: S \to T$$ be a mapping.

Then:
 * $$I_T \circ f = f$$

where $$I_T$$ is the identity mapping on $$T$$, and $$\circ$$ signifies composition of mappings.

Proof
We use the definition of relation equality, as follows:

Equality of Domains
The domains of $$\mathcal R$$ and $$I_T \circ \mathcal R$$ are equal from Domain of Composite Relation:


 * $$\operatorname{Dom} \left({I_T \circ \mathcal R}\right) = \operatorname{Dom} \left({\mathcal R}\right)$$

Equality of Codomains
The codomains of $$\mathcal R$$ and $$\mathcal R \circ I_S$$ are also easily shown to be equal.

From Codomain of Composite Relation, the codomains of $$I_T \circ \mathcal R$$ and $$I_T$$ are both equal to $$T$$.

But from the definition of the identity mapping, the codomain of $$I_T$$ is $$\operatorname{Dom} \left({I_T}\right) = T$$

Equality of Relations
The composite of $\mathcal R$ and $I_T$ is defined as:


 * $$I_T \circ \mathcal R = \left\{{\left({x, z}\right) \in S \times T: \exists y \in T: \left({x, y}\right) \in \mathcal R \and \left({y, z}\right) \in I_T}\right\}$$

But by definition of the identity mapping on $$T$$, we have that:
 * $$\left({y, z}\right) \in I_T \implies y = z$$

Hence:
 * $$I_T \circ \mathcal R = \left\{{\left({x, y}\right) \in S \times T: \exists y \in T: \left({x, y}\right) \in \mathcal R \and \left({y, y}\right) \in I_T}\right\}$$

But as $$\forall y \in T: \left({y, y}\right) \in I_T$$, this means:
 * $$I_T \circ \mathcal R = \left\{{\left({x, y}\right) \in S \times T: \left({x, y}\right) \in \mathcal R}\right\}$$

That is:
 * $$I_T \circ \mathcal R = \mathcal R$$

Hence the result.

Proof of Corollary
As a mapping is by definition also a relation, this result also holds for a mapping directly.

Also see

 * Identity Mapping is Right Identity