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Bopanna and N Sriram Balaji had lost to the French duo of Edouard Roger-Vasselin and Gael Monfils in the men's doubles first round.
It was the first time in 13 years that the 35-year-old Monfils beat the No.1-ranked player in the world.
Svitolina, who broke the psychological barrier of reaching a Grand Slam semifinal at Wimbledon in July, became the first Ukrainian to make the last four of the US Open with a 6-4, 6-4 win over Britain's Johanna Konta.
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