Real Inverse Hyperbolic Sine Function is Bijection

Theorem
The real inverse hyperbolic sine is a bijection.

Proof
From Hyperbolic Sine is Bijection over Reals and by definition of bijection, we have that $\sinh$ admits an inverse function over $\R$.

From:
 * Domain of Bijection is Codomain of Inverse
 * Codomain of Bijection is Domain of Inverse

the domain and image of hyperbolic sine over $\R$, is $\R$.