Is there a math function to find an element in a vector? I would like to write mathematically, if possible, the following statement:

Given a vector $x=[1,4,5,3]$ and an integer $j=3$, find the position of $j$ in $x$?

How to write this mathematically? 
If I am looking for the position of the minimum value in $x$, I would achieve this by $\arg\min x$. 
I guess $j^*=\operatorname{arg\,find} (x=j)$ but $\LaTeX$ does not recognize this.

If you consider the vector $x$ as a function from $[1, n]$ to $\mathbb{N}$, you can use the inverse $x^{-1}$. See https://en.wikipedia.org/wiki/Inverse_function#Preimages.
In your example, if $x=(1,4,5,3)$ then $x^{-1} (\lbrace 3 \rbrace) = \lbrace 4 \rbrace$.
As Jonathan Gafar remarked, the inverse set might contain more than one element. Then you can use a minimum to get the first one.
For example, if $x=(1,2,5,7,2)$ then $x^{-1} (\lbrace 2\rbrace)= \lbrace 2, 5 \rbrace$ and $\min x^{-1} (\lbrace 2\rbrace) = 2$.
However, I don't recommend to use this notation without a proper introduction or definition, since the inverse symbol might be ambiguous.
