Why min AC = min SRAC at the minima of AC curve? At the lowest point of long run average cost curve AC, the SAC is also at its minimum and that is not the case with any other SAC curve bounded by the envelope. Why is that?
Short-run implies that some decision variable cannot be free set, it is fixed for a time (in the short-run). In the long-run all variables may be freely set. Let us denote the fixed variable by $x$. This can take different values, so there is a family of short run cost functions is $SC(x,q)$.
Given a value $q$ let us denote the cost minimizing value of $x$ by $x^*$, i.e.
$$
x^* = \arg\min_x SC(x,q). 
$$
Cost minimization implies that in the long-run this will be the chosen value of $x$, thus
$$
C(q) = SC(x^*,q) = \min_x SC(x,q).
$$
Dividing by $q$ we get
$$
AC(q) = SAC(x^*,q) = \min_x SAC(x,q).
$$
The above holds for any $q$.