St. Bede the Venerable (672-735), who is declared Doctor of the Church by Pope Leo III (1899), and venerated in a Feast for the Catholic Church on May 25 and for the Eastern Orthodox on May 27, speaks to the universal primacy of the Pope of Rome. Notice in the first citation, Bede recognizes the invisible principle of unity, the Holy Spirit, but then admits its co-operation with the visible principle of unity, i.e. the Pope. In the second citation, Bede records the decease of Pope St. Gregory the Great, and notes that he held “pontifcial power” over all the churches of the world, not least England or the West.
“Following the example of the blessed ever-Virgin Mary, who was married and at the same time unstained, the Church conceives us as a Virgin by the working of the Holy Spirit; she gives birth to us as a Virgin without birth pangs; and as a woman married to one person but impregnated by another, throughout her individual parts that make her one and catholic, she remain visibly united to the legitimate [Roman] Pontiff set over her, but she increases in number by the invisible power of the Holy Spirit” (In Lucam; PL 92, 330B)
“At this time, that is, in the year of our Lord the blessed Pope Gregory, after having most gloriously governed the Roman Apostolic see thirteen years, six months, and ten days, died, and was translated to an eternal abode in the kingdom of Heaven. Of whom, seeing that by his zeal he converted our nation, the English, from the power of Satan to the faith of Christ, it behooves us to discourse more at large in our Ecclesiastical History, for we may rightly, nay, we must, call him our apostle; because, as soon as he began to wield the pontifical power over all the world, and was placed over the Churches long before converted [which were] to the true faith, he made our nation, till then enslaved to idols, the Church of Christ, so that concerning him we may use those words of the Apostle; “if he be not an apostle to others, yet doubtless he is to us; for the seal of his apostleship are we in the Lord.” (Ecclesiastical History, Book II, Ch. 1)
As for “over all the world” – We know it doesn’t mean “over all the Western world” because of the same wording of “all the world” in other places which include Egypt and Greece (Book 1, Ch. XXV)