Prove a property of polynomial map without using Zariski topology

Question

I want to prove the following property of polynomial map:

If $\varphi:V\to W$ is a polynomial map of affine varieties $V\subset \mathbb{A}^n,W\subset \mathbb{A}^m$ and $X\subset W$ is an algebraic subset of $W$, suppose $X\subset \varphi(V)$, then we have $\varphi^{-1}(X)$ is also an algebraic subset of $V$.

This is a well-known property, and you can also regard it as the exercise 2.7 from Fulton's Curve Book. I've search it on Google, every lecture notes prove it as a simple consequence of properties in Zariski topology. But up to now Fulton not yet introduce the concept of Zariski topology. So I'm asking that is there a proof of this simple property without using Zariski topology?

Answer 1

Suppose $X$ is the vanishing locus of the polynomials $f_1, \dots, f_n$ on $W$. Then $\varphi^{-1}(X)$ is the vanishing locus of the polynomials $f_1 \circ \varphi, \dots , f_n \circ \varphi$ on $V$, and hence is algebraic too.