Proof check from basic set theory

Question

I wonder if my proof is detailed enough. $f:X\to Y.$ To be proved: $f^{-1}(\bigcap_{\alpha}E{\alpha})=\bigcap_{\alpha}(f^{-1}E_{\alpha}) $ $$\,$$ So, here goes $$f^{-1}(\bigcap_{\alpha}E_{\alpha})\\=\{x\in X:f(x)\in \bigcap_{\alpha}E_{\alpha}\} \\=\{x\in X:f(x)\in E_{\alpha}\,\, \forall \alpha\}\\=\bigcap_{\alpha}\{x\in X:f(x)\in E_{\alpha}\}\\ =\bigcap_{\alpha}f^{-1}(E_{\alpha}) $$

The other way of showing this is to show that each is a subset of other.

Answer 1

This proof looks rigorous enough!