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.

In my opinion you should use plain text or perhaps define such a function yourself if you need it frequently. Of course, if you have plain-text definition, you can still add a formula, if you feel that it will help the potential reader.

We define $\operatorname{arg\,find}(v,\alpha)$ to be the index of the first occurence of $\alpha$ in $v$ or to be equal to $\operatorname{length}(v)+1$ if there is no such index.
  $$\operatorname{arg\,find}(v,\alpha)=\min\{i \in \mathbb{N}^+\mid i>\operatorname{length}(v)\lor v_i=\alpha\}$$
  where $v_i$ denotes the $i$-th coordinate of the vector $v$.

I hope this helps $\ddot\smile$
