Characterization of Set Equals Union of Sets

Theorem
Let $A$ be a set.

Let $\BB$ be a set of sets.

Then $A = \ds \bigcup \BB$ :
 * $\forall a \in A : \exists B \in \BB : a \in B$
 * $\forall B \in \BB : B \subseteq A$

Necessary Condition
Let $A = \ds \bigcup \BB$.

By definition of set union:
 * $\forall a \in A = \ds \bigcup \BB : \exists B \in \BB : a \in B$

From Set is Subset of Union:
 * $\forall B \in \BB : B \subseteq \ds \bigcup \BB = A$

Sufficient Condition
Let:
 * $\forall a \in A : \exists B \in \BB : a \in B$
 * $\forall B \in \BB : B \subseteq A$

From set union
 * $\forall a \in A : a \in \bigcup \BB$

By definition of subset:
 * $A \subseteq \bigcup \BB$

From Union of Subsets is Subset:
 * $\bigcup \BB \subseteq A$

By definition of set equality:
 * $A = \bigcup \BB$