Identity Mapping is Left Identity

Theorem
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 can use the definition of mapping equality.


 * The domains of $$f$$ and $$I_T \circ f$$ are equal from Domain of Composite Relation:

$$\mathrm{Dom} \left({I_T \circ f}\right) = \mathrm{Dom} \left({f}\right)$$.


 * The ranges of $$f$$ and $$f \circ I_S$$ are also easily shown to be equal.

From Range of Composite Relation, $$\mathrm{Rng} \left({I_T \circ f}\right) = \mathrm{Rng} \left({I_T}\right) = T$$.

But from the definition of the identity mapping, $$\mathrm{Rng} \left({I_T}\right) = \mathrm{Dom} \left({I_T}\right) = T$$.


 * Now we show that $$\forall x \in S: f \left({x}\right) = I_T \circ f \left({x}\right)$$.

Let $$x \in S$$.