Here's the moment Diddy's whole life changed ... TMZ has the first video of federal agents making their surprise arrest in an NYC hotel lobby -- potentially his final moment of freedom for at least the next decade.

The video clearly shows Diddy, clad in a black trench coat, walking into the Park Hyatt New York hotel in midtown Manhattan with his crew of several men -- but the moment he enters the lobby, law enforcement pounces, stops the mogul and separates him from his friends.

You can see from Diddy's team's reaction ... they're unsure of what's happening, walking around the hotel lobby in shock as the rapper's taken away. More on that later.

Sometime later, Diddy reappears on camera, but now his hands are cuffed behind his back as the feds escort him through the hallway and walk him into an elevator.

Next, they march him through the lobby and out the front door of the hotel in clear view of NYC pedestrians on the sidewalk.

As for Diddy's shocked friends in the lobby ... well, they weren't alone in that feeling. Diddy himself was also totally surprised.

Remember, he'd come to NYC in anticipation of the indictment ... and we know he was planning to surrender on Tuesday -- but, instead, the feds showed up Monday night to make this arrest.

Video taken just moments before the arrest shows how great of a time the rapper was having in NYC -- interacting with several excited fans he ran into on the streets.

Diddy's been charged with racketeering conspiracy, sex trafficking by force, fraud or coercion and transportation to engage in prostitution.

He was denied bail this week, with the judge in his appeal saying prosecutors proved "the defendant is a danger." As a result, Diddy's in lockup at the notorious MDC Brooklyn ... a detention center in New York City with a rough reputation.

As stated earlier this week, Diddy's lawyer Marc Agnifilo, reiterated how disappointed Diddy is with "the decision to pursue what we believe is an unjust prosecution of Mr. Combs by the U.S. Attorney’s Office."

Agnifilo says Diddy's a music icon and self-made entrepreneur who has spent 30 years building his empire and working to raise a family and uplift the Black community, and insists his client's not a criminal.

He's noting Diddy's been very cooperative with the investigation ... and asking people to reserve judgment until they've seen all the facts.