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Last Updated Jul 23, 2016 4:09 PM EDT

MIAMI -- Hillary Clinton introduced her new running mate, Virginia Sen. Tim Kaine, to voters on Saturday at a rally in Miami, their first appearance together as the presumptive Democratic ticket.

>Tim Kaine addresses Clinton supporters as VP pick
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Tim Kaine addresses Clinton supporters as VP pick

Tim Kaine made his first appearance as Hillary Clinton's running mate at a rally in Miami. See Kaine's first full remarks as Clinton's VP pick.

"Next week in Philadelphia, we will offer a very different vision for our country: one that is about building bridges, not walls," Clinton said, speaking to a crowd of more than 5,000 gathered at Florida International University. "My running mate is a man who doesn't just share those values, he lives them."

After considering two dozen contenders, Clinton called the Virginia senator in Rhode Island, where he was attending a fundraiser at a converted shipyard, on Friday night to ask him to be her running mate. After Kaine accepted, her campaign announced her pick in a text message to her supporters and a tweet. At the rally here, Clinton spoke first and focused on telling Kaine's story, unknown to many Americans.


Democratic presidential candidate Hillary Clinton arrives on stage to name U.S. Sen. Tim Kaine, D-Virginia, as her vice presidential running mate during a campaign rally in Miami, Florida, July 23, 2016.

Reuters/Brian Snyder

Both of the Democrats were lawyers before they entered politics, and Clinton told the crowd here that after graduating from Harvard Law School, Kaine went on to be a civil rights lawyer. She said one of his first clients was a woman who was denied an apartment because she was African-American.

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"While Tim was taking on housing discrimination and homelessness," she said, "Donald Trump was denying apartments to people who were African-American."

Kaine, who switched seamlessly between English and Spanish throughout his remarks, said that "when Trump says he has your back, you better watch out."

He slammed the Republican nominee on his business record, specifically in Atlantic City, New Jersey, and for his rhetoric, contrasting it with Clinton, who he said listens rather than insults and respects American allies.

"She will always have our backs," he said, "and I know that because she knows we are stronger together."

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Kaine also noted that Saturday might have been the "first time" some in the audience had heard him speak, or even heard his name. He spoke about his upbringing in Kansas, his experience working as a missionary in Honduras, and his decision to run for local office years ago in Richmond, Virginia, where he still lives. He shared an emotional story of the shooting at Virginia Tech in 2007, which he called "the worst day" of his life.

Reflecting on his new role as Clinton's No. 2, Kaine said that he was "grateful."

"I am grateful to you, Hillary, for the trust you put in me," Kaine said. "I am grateful for the country which has given me so much...and today, like every day, I am especially grateful to my wife Anne and to my three beautiful kids, Nat, Woody and Annella. I am the luckiest man and the luckiest husband in the world."

Clinton's vice presidential pick has been praised across the aisle.

Arizona Sen. Jeff Flake, a Republican, tweeted his congratulations to his Senate colleague, calling the Virginia Democrat a "good man and a good friend."

Trying to count the ways I hate @timkaine. Drawing a blank. Congrats to a good man and a good friend.

— Jeff Flake (@JeffFlake) July 23, 2016

And in a fundraising email blasted out Saturday, President Obama also praised Kaine for his legislative and executive track record.

"Like Hillary, Tim is an optimist," Mr. Obama wrote. "But like Hillary, he is also a progressive fighter."

But there was one person unhappy with Clinton's choice: Trump, who tweeted up a storm against Kaine Saturday morning, charging that the Virginian was "owned by the banks" and slamming Kaine's previous praise for the Obama-backed Trans Pacific Partnership (TPP).

The Bernie Sanders supporters are furious with the choice of Tim Kaine, who represents the opposite of what Bernie stands for. Philly fight?

— Donald J. Trump (@realDonaldTrump) July 23, 2016

Tim Kaine is, and always has been, owned by the banks. Bernie supporters are outraged, was their last choice. Bernie fought for nothing!

— Donald J. Trump (@realDonaldTrump) July 23, 2016

Tim Kaine has been praising the Trans Pacific Partnership and has been pushing hard to get it approved. Job killer!

— Donald J. Trump (@realDonaldTrump) July 23, 2016

Trump has repeatedly called the TPP trade deal a "rape" of the United States economy.

His own running mate, Indiana Gov. Mike Pence, was a strong backer of TPP while serving in Congress.

CBS News' Reena Flores contributed to this report.

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