Mapping from Quotient Set when Defined is Unique

Theorem
Let $S$ and $T$ be sets.

Let $\RR$ be an equivalence relation on $S$.

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

Let $S / \RR$ be the quotient set of $S$ induced by $\RR$.

Let $q_\RR: S \to S / \RR$ be the quotient mapping induced by $\RR$. Let the mapping $\phi: S / \RR \to T$ defined as:
 * $\phi \circ q_\RR = f$

be well-defined.

Then $\phi$ is unique.

Proof
From Condition for Mapping from Quotient Set to be Well-Defined, $\phi$ is well-defined :
 * $\forall x, y \in S: \tuple {x, y} \in \RR \implies \map f x = \map f y$

From Quotient Mapping is Surjection, $q_\RR$ is a surjection.

Suppose $\psi: S / \RR \to T$ is another well-defined mapping defined as:
 * $\psi \circ q_\RR = f$

Then we have:

Hence the result.

Also see

 * Definition:Well-Defined Mapping


 * Condition for Mapping from Quotient Set to be Well-Defined
 * Condition for Mapping from Quotient Set to be Injection
 * Condition for Mapping from Quotient Set to be Surjection