Consider a rational preference relation R. Prove that if u(x)=u(y) implies x~y, u(x)>u(y) implies xPy (which means x is strictly preferred to y), the u(.) is a utility function representing R.

My consideration of this problem:

for any x, if there is and only a unique u(x) exist, that is defined as the reflaction of f:x->u(x), to prove this and we can get the existence of UF representing R.

assuming that for a single x, exist more than one u(x) reflecting in the space U, say u(x) and u(y).

if u(x)=u(y), x only have one reflection in U.

if u(x)not=u(y), we can assume either u(x)>u(y) or the opposite. then from above condition we know that this must implies xPy relation. since u(y) also represent x, follows x~y. Here we generate contradiction from our assumption which means the assumption is wrong.

as a result, for a single x, there is only one u exists in the U space, which shows the function relationship between x and u(x).