Preimage of Intersection 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:

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

Proof

$f$ is a mapping.

Therefore it is by definition a many-to-one relation.

It follows from Inverse of Many-to-One Relation is One-to-Many that its inverse $f^{-1}$ is a one-to-many relation.

Thus Image of Intersection under One-to-Many Relation: General Result applies:

$\ds \RR \sqbrk {\bigcap \mathbb T} = \bigcap_{X \mathop \in \mathbb T} \RR \sqbrk X$

where here $\RR = f^{-1}$.

$\blacksquare$

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