Union of Set of Sets when a Set Intersects All

Theorem
Let $F$ be a set of sets.

Let $S$ be a set or class.

Suppose that:
 * $\forall A \in F: A \cap S \ne \O$

Then:
 * $\ds F = \bigcup_{x \mathop \in S} \set {A \in F: x \in A}$

Proof
Suppose that $B \in F$.

Then $B \cap S$ has an element $x_B$.

Thus:
 * $B \in \set {A \in F: x_B \in A}$

By the definition of union:


 * $\ds B \in \bigcup_{x \mathop \in S} \set {A \in F: x \in A}$

Suppose instead that:
 * $\ds B \in \bigcup_{x \mathop \in S} \set {A \in F: x \in A}$

Then by the definition of union, there exists an $x_B \in S$ such that:
 * $B \in \set {A \in F: x_B \in A} \subseteq F$

Thus $B \in F$.

We have shown that:
 * $\ds \forall B: \paren {B \in F \iff B \in \bigcup_{x \mathop \in S} \set {A \in F: x \in A} }$

Therefore by definition of set equality:
 * $\ds F = \bigcup_{x \mathop \in S} \set {A \in F: x \in A}$