ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4055 on D467: Injective map

Singleton map is injection from set to power set

Formulation 0

Let $X$ be a D11: Set such that

(i)	$\mathcal{P}(X)$ is the D80: Power set of $X$

Let $f : X \to \mathcal{P}(X)$ be a D18: Map such that

(i)	\begin{equation} f(x) : = \{ x \} \end{equation}

Then $f$ is an D467: Injective map from $X$ to $Y$.

Proofs

Proof 0

Let $X$ be a D11: Set such that

(i)	$\mathcal{P}(X)$ is the D80: Power set of $X$

Let $f : X \to \mathcal{P}(X)$ be a D18: Map such that

(i)	\begin{equation} f(x) : = \{ x \} \end{equation}

Since $\{ \{ x \} : x \in X \} \subseteq \mathcal{P}(X)$, this result is a consequence of the results

(i)	R2764: Canonical singleton map is bijection
(ii)	R4056: Extending codomain preserves injectivity

$\square$