Mapping Images are Disjoint only if Domains are Disjoint

Theorem
Let $S$ and $T$ be sets.

Let:
 * $f \left[{S}\right] \cap f \left[{T}\right] = \varnothing$

where $f \left[{S}\right]$ denotes the image set of $S$.

Then:
 * $S \cap T = \varnothing$

Proof
From Mapping Image of Intersection:
 * $f \left[{S \cap T}\right] \subseteq f \left[{S}\right] \cap f \left[{T}\right]$

From Empty Set is Subset of All Sets:
 * $f \left[{S \cap T}\right] = \varnothing$

From :
 * $S \cap T = \varnothing$