In the statement Osborne's mother, sister and nephew said: "We are massively in shock".
Eleven people were injured in the attack, which took place early Monday near Finsbury Park mosque, north London, raising fears of retaliation against Muslims after recent assaults by Islamic extremists.
"Hatred and evil of this kind will never succeed", she said.
"While this appears to be an attack on a particular community, like the awful attacks in Manchester, Westminster and London Bridge, it is also an assault on all our shared values of tolerance, freedom and respect", Khan said, adding that there would be "zero tolerance" for hate crimes. Two others were treated for minor injuries at the scene.
Police are holding Darren Osborne, 47, of Cardiff, Wales, on suspicion of attempted murder and terrorism offences.
As police searched a property, five residents speaking to journalists from the Press Association news agency identified images of the arrested man as their neighbour, Osborne.
Ms Sharazi said she believed the comment was not meant with ill intent and was the first conflict they had encountered with their neighbour.
The BBC was among other media which identified the man arrested as Osborne, while London's Metropolitan Police said it would not name the suspect until he was charged. Hours after the London attack, a man in Paris drove his auto into a police auto; only the attacker died, but it underscored the difficulty of defending against violence by vehicle.
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The van used in the attack is undergoing forensic examination by the police.
The victims were all Muslims and the incident was declared a terror attack on the Muslim population. Her son, Nadeem, 12, said, "I was on my bike and he just came up to me and said, 'Inbred.' Just out there [on the road]".
"People of this community have a very good working experience with the local police, they will be seeing more police in the area - particularly around religious establishments, they will see more officers on the streets, they may see more armed officers on the streets as well".
"He has a fracture in his skull - but they still don't know why the bleeding from his ear is not stopping yet", the Somali-born man said.
He said: "I couldn t believe when I realised it was him who was accused".
"This is a complete shock, I thought he was a normal neighbour, laughing and joking".
Scotland Yard said extra police patrols would be deployed to reassure the public and those observing Ramadan.
Security Minister Ben Wallace confirmed the suspect was not known to the security services.
Flowers have been laid near the scene, with one card reading: "This is an attack on all Londoners and on my community".
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