Surjective Restriction of Real Exponential Function

Theorem
Let $\exp: \R \to \R$ be the exponential function:
 * $\map \exp x = e^x$

Then the restriction of the codomain of $\exp$ to the strictly positive real numbers:
 * $\exp: \R \to \R_{>0}$

is a surjective restriction.

Hence:
 * $\exp: \R \to \R_{>0}$

is a bijection.

Proof
We have Exponential on Real Numbers is Injection.

Let $y \in \R_{> 0}$.

Then $\exists x \in \R: x = \map \ln y$

That is:
 * $\exp x = y$

and so $\exp: \R \to \R_{>0}$ is a surjection.

Hence the result.