Vanishing Ideal of Larger Subset of Affine Space is Smaller

Theorem
Let $k$ be a field.

Let $n \ge 1$ be a natural number.

Let $\mathbb A^n_k$ be the standard affine space over $k$.

Let $S \subseteq T \subseteq \mathbb A^n_k$.

Then:
 * $\map I S \supseteq \map I T$

where $\map I S$ and $\map I T$ denote the vanishing ideals of $S$ and $T$, respectively.

Proof
Let $f \in \map I T$.

Then $\map f x = 0$ for all $x \in T$.

Since $S \subseteq T$, it follows that $\map f x = 0$ for all $x \in S$.

That is, $f \in \map I S$.