Image of Intersection is Subset of Intersection of Images
Image of Intersection is Subset of Intersection of Images
For \(f : X \to Y\) and \(A_i \subseteq X\) for all \(i \in I\)
\[ f\left(\bigcap_{i \in I} A_{i}\right) \subseteq \bigcap_{i \in I} f(A_{i})\]
and in particular
\[ f\left(\bigcap_{i \in I} A_{i}\right) = \bigcap_{i \in I} f(A_{i}) \quad \text{for any collection of}\ A_i \iff f \ \text{is injective} \]