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

Result R354 on D528: Map image

Image of singleton set

Formulation 0

Let $f : X \to Y$ be a D18: Map such that

(i)	$X \neq \emptyset$
(ii)	$x \in X$

Then \begin{equation} f(\{ x \}) = \{ f(x) \} \end{equation}

Formulation 1

Let $f : X \to Y$ be a D18: Map such that

(i)	$X \neq \emptyset$
(ii)	$x \in X$

Then \begin{equation} f \{ x \} = \{ f(x) \} \end{equation}

Proofs

Proof 0

Let $f : X \to Y$ be a D18: Map such that

(i)	$X \neq \emptyset$
(ii)	$x \in X$

Directly from the definitions \begin{equation} f(\{ x \}) = \{ f(y) \mid y \in \{ x \} \} = \{ f(x) \} \end{equation} $\square$