I want to show that \(\displaystyle f(A\backslash B) \subset f(A) \backslash f(B)\)

I have done the following:

Assume that \(\displaystyle y \in f(A\backslash B)\).

This means that \(\displaystyle \exists x\in A\backslash B: f(x)=y\) (by definition).

This means that \(\displaystyle x\in A \land x\notin B\).

This means that \(\displaystyle f(x) \in f(A) \land f(x) \notin B\).

This means that \(\displaystyle f(x) \in f(A) \backslash f(B)\)

Therefore \(\displaystyle f(A\backslash B) \subset f(A) \backslash f(B)\).

Is the above correct?