Your family really comes from village Verhnyakivtsi in Borschiv district Ternopil region. Verhnyakivtsi - neighboring village of Borschiv. Your family is mixed, ie Polish and Ukrainian.
Michal Tarnavsky born January 3, 1852 in house number 69 in Verhnyakivtsi. Roman Catholic. His parents, Paul Tarnavsky and Mary Shalatska. In 1876 Michael married with Mary Humeniuk. She was born in 1859. Greek Catholic. Her parents Dmytro Humeniuk and Paraskeva Tyrka.
Michael and Mary lived in Verhnyakivtsi in house number 37 and had children (daughters baptized by the Greek Catholic rite): - Xenia - was born in 1881 (fate unknown) - Anna - born 1883 (not 1884) - Evdokia - born 1887 (not 1888) - Thekla - born and died in 1892 - Mary - born 1897 -! Joseph - for your version was born in 1883. It is probably Roman Catholic. Check information impossible as Roman Catholic metrics starting with 1884.
Surname Tarnavsky in Verhnyakivtsi was very common. In metrics is info about siblings Michael and Mary. If you need additional information please contact the family. My e-mail: firstname.lastname@example.org Igor
P.S. I've been doing some research in Verhnyakivtsi. On my page on FB you can watch Verhnyakivtsi: https://www.facebook.com/media/set/?set=a.339834182802884.78299.259994807453489&type=3