Let [tex]A[/tex], [tex]B[/tex] be sets, [tex]C,D\subset A[/tex] and [tex]f:A\longrightarrow B[/tex] be a function between them. Then [tex]f(C\cap D)=f(C)\cap f(D)[/tex] if and only if [tex]f[/tex] is injective.
Another proposition, that I have proven that for any function [tex]f(C\cap D)\subset f(C)\cap f(D)[/tex], and the definition of injectiveness: f is inyective if [tex]\forall b\in B\mid b=f(x)=f(y)[/tex] for some [tex]x,y\in A[/tex] implies that [tex]x=y[/tex].
The Attempt at a Solution
If we suppose the injectiveness is trivial to get the equality. But for the other direction I get stuck in what way to use the equality of images to get inyection. I can't see how to make a proof, in fact I can't associate the equality with the fact that there must be a unique preimage for every [tex]b\in B[/tex].