Convert GPS coordinates to Web Mercator EPSG:3857 using python/pyproj

I’ve created a couple of functions using the math package from python 3.5 to transform GPS coordinates to x/y web mercator (pseudo mercator) as explained here: http://wiki.openstreetmap.org/wiki/Mercator#Java

I took the Java version and ported it to Python:

import math

# derived from the Java version explained here: http://wiki.openstreetmap.org/wiki/Mercator
RADIUS = 6378137.0 # in meters on the equator

def lat2y(a):
  return math.log(math.tan(math.pi / 4 + math.radians(a) / 2)) * RADIUS

def lon2x(a):
  return math.radians(a) * RADIUS

Then I can use them to get for example the projection of Trafalgar Square (taking the GPS coordinates from the Google Maps URL):

ts_gm = [51.50809,-0.1285907]
print('latitude web mercator y: {} longitude web mercator x: {}'.format(lat2y(ts_gm[0]), lon2x(ts_gm[1])))

Which gives back:

latitude web mercator y: 6711665.883938471 longitude web mercator x: -14314.651244750603

However I would like to achieve this using pyproj syntax, but I am lost in the details of the string I should pass to build a projection using Web Mercator EPSG:3857.
I suspect the formulas above have some sort of precision issue and I would like to rely on a library like pyproj hoping to minimise projection errors.

Is there a minimum working example I could see to achieve the mapping in python using pyproj?

from pyproj import Proj, transform
print(transform(Proj(init='epsg:4326'), Proj(init='epsg:3857'), -0.1285907, 51.50809))  # longitude first, latitude second.
# output (meters east of 0, meters north of 0): (-14314.651244750548, 6711665.883938471)

from pyproj import Transformer


TRAN_4326_TO_3857 = Transformer.from_crs("EPSG:4326", "EPSG:3857")

def mytransform(lon, lat):
  return TRAN_4326_TO_3857.transform(lon, lat)