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:

A computer search can verify solutions under $10^6$ (that is, with no more than $6$ digits) in seconds.

Let $n$ be a $k$-digit number, for $k \ge 7$.

Then the sum of the subfactorials of its digits is not more than $\mathop !9 \times k$.

But we have:

So no more numbers have this property.