Image of Intersection under Mapping

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Let $S_1$ and $S_2$ be subsets of $S$.

Then:


 * $f \sqbrk {S_1 \cap S_2} \subseteq f \sqbrk {S_1} \cap f \sqbrk {S_2}$

This can be expressed in the language and notation of direct image mappings as:
 * $\forall S_1, S_2 \in \powerset S: \map {f^\to} {S_1 \cap S_2} \subseteq \map {f^\to} {S_1} \cap \map {f^\to} {S_2}$

That is, the image of the intersection of subsets of a mapping is a subset of the intersection of their images.

Also see

 * Image of Intersection under Injection: equality happens $f$ is an injection
 * Image of Intersection under One-to-Many Relation


 * Preimage of Intersection under Mapping


 * Image of Union under Mapping
 * Preimage of Union under Mapping