Performing logical reasoning to find minimum function between two functions

Consider the following two expressions:

f[x_,y_] := (1+x)
g[x_,y_] := (1+x*y^2)

x,y are bounded in a unit hypercube. (Ignore the fact that f[x,y] does not have y-dependence, this is a toy example). I want to find

expr=Min[f[x,y],g[x,y]]

Now in this case, g[x,y] will always give the minimum i.e. one can remove the minimum condition

expr=Min[f[x,y],g[x,y]] == g[x,y]

Question: Is there any way to achieve this i.e. to know g[x,y] is always the minimum?

What Mathematica can do, is find the range for (x,y) for which each of these functions are minimum or can also find the minimum value for each of these functions, eg. using FindMinimum, FindMinValue. The minimum value for these two functions is 1 but not necessarily for the same values of (x,y) eg. g[x,y] can give the minimum 1 even when x!=0 & y=0. Also whenever f[x,y] gives minimum (for x=0), this automatically enforces g[x,y] to be minimum, leading to the deduction that expr=g[x,y]. Any suggestion?