Intersection of Images of a function

Intuitively this is because a mapping $f:A\rightarrow B$ may identify elements in the sense of:

$$f(x)=f(y) \text{ in } B \text{ where } x\neq y \text{ in }A$$

but a mapping by definition may not distinguish or separate an element into two in this sense:

$$f(x)\neq f(y)\text{ in } B \text{ where } x=y \text{ in } A$$

Say $A_1, A_2\subset A$. Think of $f(A_1)\cap f(A_2)$ as the "identifications" done by $f$ in $B$ of elements from $A$.

Then the mapping $f$ necessarily "identifies" all elements from $A_1\cap A_2$ - i.e.: $$f(A_1\cap A_2)\subset f(A_1)\cap f(A_2)$$

All that can happen additionally during $f$ mapping is that there are more points from $A_1$ and $A_2$ that are not from $A_1\cap A_2$ that are identified by $f$ in $B$ in which case we have $$f(A_1\cap A_2)\not\supset f(A_1)\cap f(A_2)$$