$f:X\to Y$ is an open map. Fix $x\in X$. Why is $\bigcap_{x\in U\in T_1}{}^*f({}^*U)\subseteq {}^*f(\bigcap_{x\in U\in T_1}{}^*U)$ true?

Question

Suppose that $f:X\to Y$ is an open map. Fix $x\in X$. Why is $\bigcap_{x\in U\in T_1}{}^*f({}^*U)\subseteq {}^*f(\bigcap_{x\in U\in T_1}{}^*U)$ true?

https://iugspace.iugaza.edu.ps/bitstream/handle/20.500.12358/21375/file_1.pdf?sequence=1&isAllowed=y

Theorem 2.5.5

I have tried several ways, such as considering the infinitesimal open neighbor hood of $x$, or constructing a well-ordered sequence of its open neighbor hoods and trying to show that if $\alpha$ is an ordinal number then there can not be a minimal $\beta\in{}^*\alpha$ such that ${}^*\gamma<\beta$ holds for every $\gamma<\alpha$.

None of them works. Please help me.