Image of Union under Mapping/General Result

Theorem

Let $S$ and $T$ be sets.

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

Let $\powerset S$ be the power set of $S$.

Let $\mathbb S \subseteq \powerset S$.

Then:

$\ds f \sqbrk {\bigcup \mathbb S} = \bigcup_{X \mathop \in \mathbb S} f \sqbrk X$

This can be expressed in the language and notation of direct image mappings as:

$\ds \forall \mathbb S \subseteq \powerset S : \map {f^\to} {\bigcup \mathbb S} = \bigcup_{X \mathop \in \mathbb S} \map {f^\to} X$

Proof

As $f$, being a mapping, is also a relation, we can apply Image of Union under Relation: General Result:

$\ds \RR \sqbrk {\bigcup \mathbb S} = \bigcup_{X \mathop \in \mathbb S} \RR \sqbrk X$

$\blacksquare$

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Functions
• 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions: Exercise $7 \ \text {(a)}$
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions: Exercise $2.3$