Number which is Sum of Subfactorials of Digits

Theorem
The only integer which is the sum of the subfactorials of its digits is $148 \, 349$:
 * $148 \, 349 = \mathop !1 + \mathop !4 + \mathop !8 + \mathop !3 \mathop + \mathop !4 \mathop + \mathop !9$

Proof
We have: