Strictly Monotone Real Function is Bijective

Theorem
Let $$f$$ be a real function which is defined on $$I \subseteq \R$$.

Let $$f$$ be strictly monotone on $$I$$.

Let the image of $$f$$ be $$J$$.

Then $$f: I \to J$$ is a bijection.

Proof
From Strictly Monotone Mapping is Injective, $$f$$ is an injection.

From Surjection by Restriction of Range, $$f: I \to J$$ is a surjection.