Identity Mapping is Right Identity

Theorem

Let $S$ and $T$ be sets.

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

Then:

$f \circ I_S = f$

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

The codomains of $f$ and $f \circ I_S$ are both equal to $T$ from Codomain of Composite Relation.

$\Box$

$\Dom {f \circ I_S} = \Dom {I_S}$

But from the definition of the identity mapping:

$\Dom {I_S} = \Img {I_S} = S$

$\Box$

The composite of $I_S$ and $f$ is defined as:

$f \circ I_S = \set {\tuple {x, z} \in S \times T: \exists y \in S: \tuple {x, y} \in I_S \land \tuple {y, z} \in f}$

But by definition of the identity mapping on $S$, we have that:

$\tuple {x, y} \in I_S \implies x = y$

Hence:

$f \circ I_S = \set {\tuple {y, z} \in S \times T: \exists y \in S: \tuple {y, y}\ \in I_S \land \tuple {y, z} \in f}$

But as $\forall y \in S: \tuple {y, y} \in I_S$, this means:

$f \circ I_S = \set {\tuple {y, z} \in S \times T: \tuple {y, z} \in f}$

That is:

$f \circ I_S = f$

$\Box$

Hence the result, by Equality of Mappings.

$\blacksquare$

By definition, a mapping is also a relation.

Also by definition, the identity mapping is the same as the diagonal relation.

$\blacksquare$

1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 2$: Product sets, mappings
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 5$: Composites and Inverses of Functions
1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 16$
1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.14$: Composition of Functions: Theorem $14.7$
1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Definition $2.11$
1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.5$: Identity, One-one, and Onto Functions
2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.2$