The injective image of intersections is the intersection of images

up:: The image of the intersection is contained in the intersection of the images

When $f:X\to Y$ is an Injective Function, we have, in particular, that¹

$$(\exists x\in A\mid f(x)=y)\wedge (\exists z\in B\mid f(z)=y)⟹(x=z)⟹\exists x\in A\cap B\mid f(x)=y$$

thus proving the reciprocal $f(A)\cap f(B)\subseteq f(A\cap B)$.

Footnotes

1. That is due to $\forall y\in Y,\exists !x\in X\mid f(x)=y$. ↩