elementary set theory - Is this proof correct for : Does $F(A)cap F(B)subseteq F(Acap B)$ for all functions $F$?

Is this proof correct? To prove $F(A)\cap F(B)\subseteq F(A\cap B)$ for all functions $F$.

Let any number $y\in F(A)\cap F(B)$. We want to show $y\in F(A\cap B).$

Therefore, $y\in F(A)$ and $y\in F(B)$, by definition of intersection.

By definition of inverse, $y=F(x)$ for some $x\in A$ and $x\in B$

And so, $y=F(x)$ for some $x\in A\cap B$

And therefore, $y\in F(A\cap B)$

I have a gut feeling deep down that something is wrong. Can anyone help me pinpoint the mistake? I am not sure why am I having so much problems with functions. Am I not thinking in the right direction?

Sources : 2nd Ed, P219 9.60 = 3rd Ed, P235 9.12, 9.29 - Mathematical Proofs, by Gary Chartrand,
P214 Theorem 12.4 - Book of Proof, by Richard Hammack,
P257-258 - How to Prove It, by D Velleman.

Answer

The third line is mistaken. You only know that there exists an $x$ in $A$ such that $F(x)=y$, and you know there is a $z\in B$ such that $F(z)=y$.

It is extremely easy to find a counterexample: just draw two sets $A$, $B$ that are disjoint, and map an $a\in A$ and a $b\in B$ to a single point. Then you have that $y\in F(A)\cap F(B)$, but $F(A\cap B)=\emptyset$.