Image Preserves Union
Image Preserves Union
For \(f : X \to Y\) and \(A_i \subseteq X\) for all \(i \in I\)
\[ f\left(\bigcup_{i \in I} A_{i}\right) = \bigcup_{i \in I} f(A_{i})\]
Proof
\[\begin{align*}
x \in f(\cup_{i \in I} A_{i}) &\iff \exists y \in \cup_{i \in I} A_i, (f(y) = x) \\
&\iff \exists i \in I, \exists y \in A_i, (f(y) = x) \\
&\iff \exists i \in I, x \in f(A_i) \\
&\iff x \in \cup_{i \in I} f(A_i) \\
\end{align*}\]