DataFrame.set_index() may not preserve dtype
xref #19602

#### Code Sample, a copy-pastable example if possible

```python
In [1]: import pandas as pd

In [2]: pd.__version__
Out[2]: '0.25.3'

In [3]: df = pd.DataFrame({'mixed' : [1, 2, 'abc', 'def'], 'ints': [100, 200, 3
   ...: 00, 400]})

In [4]: df
Out[4]:
  mixed  ints
0     1   100
1     2   200
2   abc   300
3   def   400

In [5]: df.dtypes
Out[5]:
mixed    object
ints      int64
dtype: object

In [6]: df.query('ints < 300').set_index('mixed').index
Out[6]: Int64Index([1, 2], dtype='int64', name='mixed')

In [7]: df.set_index('mixed').query('ints < 300').index
Out[7]: Index([1, 2], dtype='object', name='mixed')

```
#### Problem description

In the above, I start with a `DataFrame` with a column `mixed` that has both integer and string values. 

In statement [6],  I do a query on a different column and then set the index to be the column `mixed`.  The resulting index now has an `int64` dtype as opposed to having the dtype preserved from the original column.  

But in statement [7], I first set the index, and then do the query, and now the index has the `object` dtype.

This becomes an issue if one does some computation on the queried DataFrame and then create the index `mixed`, and then you want to merge it back to the original `DataFrame`.  Now the original one will have `mixed` as dtype `'O'` and the new one has `mixed` as dtype '`int`'

#### Expected Output

From statement [6], I would have expected:
```
Index([1, 2], dtype='object', name='mixed')
```

#### Output of ``pd.show_versions()``

<details>

INSTALLED VERSIONS
------------------
commit           : None
python           : 3.7.3.final.0
python-bits      : 64
OS               : Windows
OS-release       : 10
machine          : AMD64
processor        : Intel64 Family 6 Model 158 Stepping 13, GenuineIntel
byteorder        : little
LC_ALL           : None
LANG             : None
LOCALE           : None.None

pandas           : 0.25.3
numpy            : 1.17.4
pytz             : 2019.3
dateutil         : 2.8.1
pip              : 19.3.1
setuptools       : 42.0.2.post20191203
Cython           : 0.29.14
pytest           : 5.3.2
hypothesis       : 4.54.2
sphinx           : 2.3.0
blosc            : None
feather          : None
xlsxwriter       : 1.2.6
lxml.etree       : 4.4.2
html5lib         : 1.0.1
pymysql          : None
psycopg2         : None
jinja2           : 2.10.3
IPython          : 7.10.2
pandas_datareader: None
bs4              : 4.8.1
bottleneck       : 1.3.1
fastparquet      : None
gcsfs            : None
lxml.etree       : 4.4.2
matplotlib       : 3.1.1
numexpr          : 2.7.0
odfpy            : None
openpyxl         : 3.0.2
pandas_gbq       : None
pyarrow          : None
pytables         : None
s3fs             : None
scipy            : 1.3.2
sqlalchemy       : 1.3.11
tables           : 3.6.1
xarray           : None
xlrd             : 1.2.0
xlwt             : 1.3.0
xlsxwriter       : 1.2.6
</details>
