Image of Intersection under Relation/Family of Sets

From ProofWiki
Jump to navigation Jump to search


Let $S$ and $T$ be sets.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.

Let $\RR \subseteq S \times T$ be a relation.


$\ds \RR \sqbrk {\bigcap_{i \mathop \in I} S_i} \subseteq \bigcap_{i \mathop \in I} \RR \sqbrk {S_i}$

where $\ds \bigcap_{i \mathop \in I} S_i$ denotes the intersection of $\family {S_i}_{i \mathop \in I}$.


\(\ds \forall i \in I: \, \) \(\ds \bigcap_{j \mathop \in I} S_j\) \(\subseteq\) \(\ds S_i\) Intersection is Subset: Family of Sets
\(\ds \leadsto \ \ \) \(\ds \forall i \in I: \, \) \(\ds \RR \sqbrk {\bigcap_{j \mathop \in I} S_j}\) \(\subseteq\) \(\ds \RR \sqbrk {S_i}\) Image of Subset is Subset of Image
\(\ds \leadsto \ \ \) \(\ds \RR \sqbrk {\bigcap_{i \mathop \in I} S_i}\) \(\subseteq\) \(\ds \bigcap_{i \mathop \in I} \RR \sqbrk {S_i}\) Intersection is Largest Subset: Family of Sets