Cardinality of Image of Mapping of Intersections is not greater than Weight of Space

Theorem
Let $T = \struct {X, \tau}$ be a topological space.

Let $f: X \to \tau$ be a mapping such that:
 * $\forall x \in X: \paren {x \in \map f x \land \forall U \in \tau: x \in U \implies \map f x \subseteq U}$

Then the cardinality of the image of $f$ is no greater than the weight of $T$:


 * $\card {\Img f} \le \map w T$

Proof
By definition of weight, there exists a basis $\BB$ of $T$ such that:
 * $\card \BB = \map w T$

By Image of Mapping of Intersections is Smallest Basis:
 * $\Img f \subseteq \BB$

Thus by Subset implies Cardinal Inequality:
 * $\card {\Img f} \le \card \BB = \map w T$