일 (一) : first 등 (等) : grade, class 일등 : first grade (http://hanmun.vdegenne.com/일/一/0/일등) 공 (功) : work 신 (臣) : to serve a ruler (e.g. 간신 is like a betrayal towards the king) 공신 : work loyal to the country / who made a nice contribution (http://hanmun.vdegenne.com/공/功/0/공신) I hope that it helps you

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'일등' means ' No.1', and '공신' means 'A king's servant(kind of government officer) who made a great, special contribution to his country/nation.' So, '일등 공신' means 'No.1 공신 among them.' Without 일등 공신, the country/nation couldn't be better than before, cause he was a key figure of it. So, '일등 공신' can be said when 'someone/something' did a key role for a better situation. (You can also say it ironically) Simply, it means 'main(major, key) reason(role).' ex) 내 건강의 일등 공신은 운동이다. 그 사람이 대통령이 될 수 있었던 이유는 다양하다. 하지만 그중 일등 공신은 그 사람의 아내이다. For that reason, '일등 공신' in your mentioned sentence means, '정기 검진' do the key role of 암 정복.