if I understand well : private key is a combining of random number between 1 and 10^77.
From this private key is derived a bitcoin address with a precise algorithm.
As long as you send BTC to this bitcoin address which has been derived from your private key : you can safely say that you have the control of your fund and that your private key is valid.
can someone confirm?
Correct.
If I recall correctly you can actually generate two bitcoin addresses from a given random number. But it's certainly correct that the process is completely specified. There's a python library that you can use to play around with keys from numbers, maybe 'ku' (key utils), don't remember. Anyway, python is convenient for learning, I think, the code is pretty readable and you can use the REPL to explore interactively.