Preimage of Union under Mapping/General Result

Theorem

Let $S$ and $T$ be sets.

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

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

Let $\mathbb T \subseteq \powerset T$.

Then:

$\ds f^{-1} \sqbrk {\bigcup \mathbb T} = \bigcup_{X \mathop \in \mathbb T} f^{-1} \sqbrk X$

Proof

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

$\ds \RR^{-1} \sqbrk {\bigcup \mathbb T} = \bigcup_{X \mathop \in \mathbb T} \RR^{-1} \sqbrk X$

$\blacksquare$

Sources

• 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 $6 \ \text {(a)}$
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 2$: Functions: Exercise $2.3$