We know by the standard Implicit Function Theorem that

If $f:\mathbb R^4\rightarrow\mathbb > R^2$ is a polynomial (or in fact any continuously differentiable function), then there is a point $a\in\mathbb > R^2$ such that $f^{-1}(a)$ is at least two-dimensional.

Now imagine that instead of $\mathbb R^2$ we have an algebraic surface $S\subset\mathbb R^3$, i.e. the zero set of a trivariate polynomial. It's reasonable that the statement still holds. A general statement along these lines would be something like

If $A,B$ are two algebraic sets such that $\dim(A)>\dim(B)$, and if $f:A\rightarrow B$ is a polynomial mapping, then there is a point $b\in > B$ such that $f^{-1}(b)$ is of dimension at least $\dim(A)-\dim(B)$.

Is it a well-known theorem? Any reference for it?