How do you show the image of the intersecction of S and T is a subset of the intersection of the image of S and the image of T if f is a function from A to B and S And T are subsets of A?

Let function f from A into B.

Let S and T are subset of A.

To show that $f\left(A\cap B\right)\subset f\left(A\right)\cap f\left(B\right)$.

Let $y\in f\left(A\cap B\right)$.

To prove that $y\in f\left(A\right)\cap f\left(B\right)$.

$y\in f\left(A\cap B\right)$  means that there exist $x\in A\cap B$such that $f\left(x\right)=y$.

Here, $x\in A\cap B$ implies that $x\in A$ such that $f\left(x\right)=y$ and $x\in B$ such that $f\left(x\right)=y$.

That is, $y\in f\left(A\right) and y\in f\left(B\right)$

Implies that $y\in f\left(A\right)\cap f\left(B\right)$.

Thus, $f\left(A\cap B\right)\subset f\left(A\right)\cap f\left(B\right)$.