Definition df-iun 2558

Description: Define indexed union. Definition of [Stoll] p. 45. In normal use, A is independent of x, and B depends on x i.e. can be read informally as B(x). We call x the index, A the index set, and B the indexed set. In most books, x e. A is written as a subscript or underneath a union symbol U.. We use a special union symbol U_ to make it easier to distinguish from plain class union. In many theorems, you will see that x and A are in the same distinct variable group (meaning A cannot depend on x) and that B and x do not share a distinct variable group (meaning that can be thought of as B(x) i.e. can be substituted with a class expression containing x). An alternate definition tying indexed union to ordinary union is dfiun2 2577. Theorem uniiun 2591 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 3850 and funiunfv 3851 are useful when B is a function.

Assertion

Ref	Expression
df-iun	|- U_x e. A B = {y | E.x e. A y e. B}

Distinct variable groups:   x,y   y,A   y,B

Detailed syntax breakdown of Definition df-iun

Step	Hyp	Ref	Expression
1		vx	. . 3 set x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	ciun 2556	. 2 class U_x e. A B
5		vy	. . . . . 6 set y
6	5	cv 952	. . . . 5 class y
7	6, 3	wcel 955	. . . 4 wff y e. B
8	7, 1, 2	wrex 1638	. . 3 wff E.x e. A y e. B
9	8, 5	cab 1456	. 2 class {y | E.x e. A y e. B}
10	4, 9	wceq 953	1 wff U_x e. A B = {y | E.x e. A y e. B}

This definition is referenced by:  eliun 2560  iunss1 2564  hbiu1 2574  dfiun2g 2576  cbviun 2579  uniiun 2591  iunun 2603  abrexex2 3856

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