The universality of the grace is grounded in the unity of God. Comp. Rom 3:30. One divine purpose for all implies one God who purposes.
One God
These Epistles deal much with the divine attributes. See 1Ti 1:17; 1Ti 6:13, 1Ti 6:15, 1Ti 6:16; 1Ti 3:15; 1Ti 4:10; 2Ti 2:13; Tit 1:2.
Mediator (μεσίτης)
See on Gal 3:19. The word twice in Paul, Gal 3:29, Gal 3:20, once of Moses and once generally. In Hebrews always of Christ; Heb 8:6; Heb 9:15, Heb 12:24. This is the only instance in the pastorals. As the one God, so the one mediator implies the extension of the saving purpose to all.