I need a little confirmation on the following proof:

The problem is this:

For an interval I, with f:I -> R where f is strictly increasing with f(I) an interval, prove that f^(-1):R -> I is strictly increasing.

My proof is this:

Because f is strictly increasing, for y,z exists in I such that y<z,

f(y)<f(z).

Let y,z exist in I. Then f(y),f(z) exist in R. and f(y) < f(z).

but f^(-1)[f(y)]<f^(-1)[f(z)] which implies y<z and thus f^(-1) is strictly increasing.

This proof seems trivial to me, which makes me think I did something wrong. I possibly assumed too much in my proof, not sure. Any ideas?