Is this how this question supposed to be solved? (Writing a system of constraints that represents this connection between variables). 
Given that $x,y,z,w,v\in \{0,1\}$. The connection between the variable is given by: 
$\max\{\min\{x,y\},z,v\}=w$. 
Write a system of linear constraints that represents this connection.

My Work: 
Let $t=\min\{x,y\}$. So our system will look like: 
$\max\{t,z,v\}=w$ 
S.t: $t=\min\{x,y\} \Longrightarrow t\le x, t\le y$. 

Now, let $r=\max\{t,z,v\}$, that means we get: 
$r=w$. 
S.t: $t\le x, t\le y, r=\max\{t,z,v\}$. 
Which is also: $t\le x, t\le y, r\ge t, r\ge z, r\ge v$. and of course $x,y,z,w,v\in \{0,1\}$
I would appreciate any feedback about my work and if this is how the solution should look like. (I've dealt with writing a new equivalent minimum/maximum problem, but not with system of constraints).
Edit: I've noticed that I can't write $t=\min\{x,y\} \Longrightarrow t\le x, t\le y$, that easily, because then $t$ could be anything lower than $0$ or $1$, so I'm thinking of adding the constraint $t\in \{0,1\}$, does that still count as a linear constraint?
 A: Your constraints are incomplete.  For example, they do not prevent $(x,y,z,v,w)=(0,0,0,0,1)$, which violates the desired equation.
For binary variables, $\max$ is the same as $\lor$ (logical OR) and $\min$ is the same as $\land$ (logical AND).
You can find linear constraints somewhat automatically, without any additional variables, via conjunctive normal form as follows:
$$
w=\max\{\min\{x,y\},z,v\} \\
w \iff ((x \land y) \lor z \lor v) \\
\left(w \implies ((x \land y) \lor z \lor v)\right)
\bigwedge 
\left(((x \land y) \lor z \lor v) \implies w\right) 
\\
\left(\lnot w \lor ((x \land y) \lor z \lor v)\right)
\bigwedge 
\left(\lnot((x \land y) \lor z \lor v) \lor w\right) 
\\
\left((\lnot w \lor x \lor z \lor v) \land (\lnot w \lor y \lor z \lor v)\right)
\bigwedge 
\left(((\lnot x \lor \lnot y) \land \lnot z \land \lnot v) \lor w\right) 
\\
(\lnot w \lor x \lor z \lor v) \bigwedge (\lnot w \lor y \lor z \lor v)
\bigwedge 
(\lnot x \lor \lnot y \lor w) \bigwedge (\lnot z \lor w) \bigwedge (\lnot v \lor w) 
\\
(1-w + x + z + v \ge 1) \bigwedge (1-w + y + z + v \ge 1)
\bigwedge 
(1-x + 1-y + w \ge 1) \bigwedge (1-z + w \ge 1) \bigwedge (1-v + w \ge 1) 
\\
(x + z + v \ge w) \bigwedge (y + z + v \ge w)
\bigwedge 
(w \ge x+y-1) \bigwedge (w \ge z) \bigwedge (w \ge v) 
$$
That is, the following five linear constraints enforce the desired behavior:
\begin{align}
x + z + v &\ge w \\
y + z + v &\ge w \\
w &\ge x+y-1 \\
w &\ge z \\
w &\ge v
\end{align}
