Renaming Mapping is Bijection/Proof 2

Theorem

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

Let $r: S / \RR_f \to \Img f$ be the renaming mapping, defined as:

$r: S / \RR_f \to \Img f: \map r {\eqclass x {\RR_f} } = \map f x$

where:

$\RR_f$ is the equivalence induced by the mapping $f$

$S / \RR_f$ is the quotient set of $S$ determined by $\RR_f$

$\eqclass x {\RR_f}$ is the equivalence class of $x$ under $\RR_f$.

By definition, $\RR_f$ is the equivalence relation induced by the mapping $f$.

Hence by definition:

$\tuple {s_1, s_2} \in \RR_f \iff \map f {s_1} = \map f {s_2}$

From Condition for Mapping from Quotient Set to be Injection, this is precisely the condition required for $r$ to be an injection.

Next it is noted that the codomain of $r$ is $\Img f$.

Then from Restriction of Mapping to Image is Surjection, we have that $f_{\restriction \Img f}$ is a surjection.

Thus $r$ is shown to be both an injection and a surjection, and so by definition is a bijection.

$\blacksquare$

1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 6$. Indexed families; partitions; equivalence relations: Theorem $6.6$