"She’s really enjoying these few days just being with the baby," a source tells PEOPLE of Kim Kardashian West

By Aili Nahas and Karen Mizoguchi
January 19, 2018 08:01 PM

Kim Kardashian West‘s lifelong dream has become a reality with the arrival of her daughter Chicago.

A source tells PEOPLE that the Keeping Up with the Kardashians reality star, 37, and husband Kanye West “love the name” they chose for their third child and are “really happy” as a family of five.

“Kim is just so thrilled that her dream of having three kids finally came true,” the source said. “She’s really enjoying these few days just being with the baby and her family and soaking it all in.”

Adding, “She was a little nervous about how Saint and North would react but so far everything is great.”

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Kim Kardashian in November 2017
Jason LaVeris/FilmMagic

An insider previously told PEOPLE that the couple’s 4½-year-old daughter and 2-year-old son have been receiving “a lot of extra attention” and “feel included” in the celebration of their little sister.

On Friday, Kardashian West finally revealed the name of their baby girl four days after she was born via surrogate on Jan. 15.

While the mother of three did not give an explanation behind the name choice, Kanye, notably, was raised in Chicago and frequently pays homage to his hometown in his songs.

Kim Kardashian and Kanye West with their kids North and Saint

Pregnant sister Khloé Kardashian, who is expecting her first childa boy – with NBA boyfriend Tristan Thompson, was among the celebrities to share their opinions on the name.

“I LOOOOOOOOOOVE her name. Hey Chi (shy),” she tweeted with the pronunciation of the newborn’s nickname.

The Chicago White Sox, the Chicago Bears and NBC’s Chicago Fire also gave shout-outs to Kardashian West’s third child.

WATCH: Kim Kardashian West Warmed to Surrogacy Because ‘She Really Misses Having a Baby in the Family’: Source

While Kim and Kanye may not have given any distinct clues ahead of the name reveal, brother Rob Kardashian may have dropped a hint. On Wednesday, he tweeted two (Chicago?) bear emojis.

One family member who may have contributed to the selection of Chicago’s name: big sister North, who was tasked with the responsibility shortly before the birth.

“Kim and Kanye are letting North help with naming the baby,” an insider divulged to PEOPLE in December.