since everything hinges on this then let's explore why carrying a mat is or is not breaking the Sabbath.
the text doesn't give us specific clues nor directly calls out the pharsisies or their twisted laws. it says in v9b,10
This happened on a Sabbath day. So the Jewish leaders said to the man who had been healed, “It is the Sabbath day. The law does not allow you to carry your mat.”
the leaders then were interested in finding out who Jesus was from this man (who later told them)
there's no doubt in my mind that Jesus knew this man would be confronted and this is one of the reasons why he told him to pick up his mat. But does it violate the Sabbath? possibly a related sabbath law could be Jer 17:21-22 which reads:
Thus says the Lord: Take care for the sake of your lives, and do not bear a burden on the Sabbath day or bring it in by the gates of Jerusalem. And do not carry a burden out of your houses on the Sabbath or do any work, but keep the Sabbath day holy, as I commanded your fathers.
so is a mat "a burden"... I don't know, I'm just thinking out loud.