Equivalence of Definitions of Real Exponential Function/Inverse of Natural Logarithm equivalent to Differential Equation

Proof
==== Inverse of Natural Logarithm implies Solution of Differential Equation ====

Let $\exp x$ be the real function defined as the inverse of the natural logarithm:
 * $y = \exp x \iff x = \ln y$

Then:

This proves that $y$ is a solution of the differential equation.

It remains to be proven that $y$ fulfils the initial condition:

That is:

$\exp x$ is the solution of the differential equation:
 * $\dfrac {\d y} {\d x} = y$

satisfying the initial condition $f \left({0}\right) = 1$.

==== Solution of Differential Equation implies Inverse of Natural Logarithm ====

Let $\exp x$ be the real function defined as the solution of the differential equation:
 * $\dfrac {\d y} {\d x} = y$

satisfying the initial condition $f \left({0}\right) = 1$.

Thus:

To solve for $C$, put $\left({x_0, y_0}\right) = \left({0, 1}\right)$:

That is:
 * $y = \exp x \iff x = \ln y$