functions - Is this a valid proof of $f(S cap T) subseteq f(S) cap f(T)$?

$f(S \cap T) \subseteq f(S) \cap f(T)$

Suppose there is a $x$ that is in $S$, but not in $T$, then there is a value $y$ such that $f(x) = x$, that is in $f(S)$, but not in $f(S \cap T)$. Suppose there is a $x$ that is in $T$, but not in $S$, then there is a value $y$ such that $f(x) = x$ that is in $f(T)$, but is not in $f(S \cap T)$.

*correction made