Assume that I have a discrete set $L$ and a transformation $\phi: L \rightarrow [0,1]$ that normalizes set $L$ such that now values belonging $L$ are uniformly distributed among the unit interval.
How do I mathematically state that $\phi(L)$ adheres to uniform distribution?
If repeated values can't occur in $L$, I think the word you're probably after is a form of probability points (used in probability plotting, something like a sample percentile rank).
If the values in $L$ are $\mathbf{x}=(x_1,x_2,...,x_n)$ and $r(L)=\text{rank}(\mathbf{x})$ (which for distinct values are equivalently the indices of the order statistics), then one definition, $p_i=\frac{r_i-\alpha}{n+1-2\alpha}$ would seem to give the kind of thing you seem to be after. Specifically, with $\alpha=1$, you would reproduce the example we discussed in comments:
This is available as a standard function in a number of stats packages. So for example, R has such a probability points function, ppoints:
(L=sort(rnorm(4)))
[1] -2.5736722 -0.1084315 0.1672061 1.4856733
ppoints(L,a=1)
[1] 0.0000000 0.3333333 0.6666667 1.0000000
ppoints(c(1,2,4,17,190),a=1)
[1] 0.00 0.25 0.50 0.75 1.00
The probability points are equispaced by construction; they're a linear transformation of the ranks, $1,2, ..., n$.
