How does $\left\{f\mid f\colon\mathbb{N}\to\mathbb{R}\right\}$ mean 'the set of all real-valued functions of one natural number variable'? On pg. 83 of Hefferon's Linear Algebra, it says this:

The set  $\left \{ f\mid f\colon\mathbb{N}\rightarrow \mathbb{R} \right \}$ of all real-valued functions of one natural number variable is a vector space under the operations.

How do I read the notation to mean what the sentence says? I thought the "|" and ":" mean the same thing--"such that...". So I thought someone would write it $\left \{ f\mid \mathbb{N}\rightarrow \mathbb{R} \right \}$. Also, I don't understand the "$\mathbb{N}\rightarrow \mathbb{R}$" part. I saw from Wikipedia that "$\rightarrow$" means "implies". It makes the notation sound like "$\mathbb{N}$ implies $\mathbb{R}$", which doesn't make sense to me.

If we read it out loud we would say:
The set of functions $f$ such that the domain of $f$ is $\mathbb N$ and the co-domain of $f$ is $\mathbb R$
