Israel

the name conferred on Jacob after the great prayer-struggle at
Peniel (Gen. 32:28), because "as a prince he had power with God
and prevailed." (See JACOB T0001945.) This is the common name
given to Jacob's descendants. The whole people of the twelve
tribes are called "Israelites," the "children of Israel" (Josh.
3:17; 7:25; Judg. 8:27; Jer. 3:21), and the "house of Israel"
(Ex. 16:31; 40:38).

This name Israel is sometimes used emphatically for the true
Israel (Ps. 73:1: Isa. 45:17; 49:3; John 1:47; Rom. 9:6; 11:26).

After the death of Saul the ten tribes arrogated to themselves
this name, as if they were the whole nation (2 Sam. 2:9, 10, 17,
28; 3:10, 17; 19:40-43), and the kings of the ten tribes were
called "kings of Israel," while the kings of the two tribes were
called "kings of Judah."

After the Exile the name Israel was assumed as designating the
entire nation.