Equivalence of Definitions of Filter Basis

Theorem
Let $S$ be a set.

Let $\FF$ be a filter on $S$.

$(1)$ implies $(2)$
Let $\BB$ be a filter basis of $\FF$ by definition $1$.

Then by definition:
 * $\BB \subset \powerset S$ such that $\O \notin \BB$ and $\BB \ne \O$

and $\FF := \set {V \subseteq S: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ :
 * $\forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$

Let $U \in \FF$.

Then by definition of $\FF$:
 * $\exists V \in \BB: V \subseteq U$

Thus $\BB$ is a filter basis of $\FF$ by definition $2$.

$(2)$ implies $(1)$
Let $\BB$ be a filter basis of $\FF$ by definition $2$.

By definition, $\BB$ is a filter basis of $\FF$ :
 * $\forall U \in \FF: \exists V \in \BB: V \subseteq U$

Let $V_1, V_2 \in \BB$.

Then:
 * $V_1, V_2 \in \FF$

By definition of filter:


 * $V_1 \cap V_2 \in \FF$

and so:
 * $\exists U \in \BB: U \subseteq V_1 \cap V_2$

Thus $\BB$ is a filter basis of $\FF$ by definition $1$.