[Math] f(X ∩ Y) = f(X) ∩ f(Y) for all non empty subsets X and Y of A, given f:A→B is 1-1

Question

Let A and B be nonempty sets and f:A→B be a 1-1 function. Then f(X ∩
Y) = f(X) ∩ f(Y) for all non empty subsets X and Y of A.

I believe this statement is true?

Answer 1

If $f:A\to B$ is not one-one, we can find $x,y$ for which $x\neq y$ yet $z=f(x)=f(y)$. Consider $X=\{x\}$ and $Y=\{y\}$. The converse is also true: if $f(X)\cap f(Y)=f(X\cap Y)$ for any $X,Y$ then $f$ is one-one.