Intersection with Subset is Subset

Theorem

 * $S \subseteq T \iff S \cap T = S$

where:
 * $S \subseteq T$ denotes that $S$ is a subset of $T$
 * $S \cap T$ denotes the intersection of $S$ and $T$.

Proof
Let $S \cap T = S$.

Then by the definition of set equality, $S \subseteq S \cap T$.

Thus:

Now let $S \subseteq T$.

From Intersection is Subset we have $S \supseteq S \cap T$.

We also have:

So as we have:

it follows from the definition of set equality that:
 * $S \cap T = S$

So we have:

and so:
 * $S \subseteq T \iff S \cap T = S$

from the definition of equivalence.

Also see

 * Union with Superset is Superset