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} $\square$