Definition df-iun 3823

Description: Define indexed union. Definition indexed union in [Stoll] p. 45. In most applications, A is independent of x (although this is not required by the definition), 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 disjoint variable group (meaning A cannot depend on x) and that B and x do not share a disjoint 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 3855. Theorem uniiun 3874 provides a definition of ordinary union in terms of indexed union. (Contributed by NM, 27-Jun-1998.)

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

Allowed substitution hints:   A( x)   B( x)

Detailed syntax breakdown of Definition df-iun

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	ciun 3821	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1331	. . . . 5 class y
7	6, 3	wcel 1481	. . . 4 wff y e. B
8	7, 1, 2	wrex 2418	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2126	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1332	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3825  nfiunxy  3847  nfiunya  3849  nfiu1  3851  dfiunv2  3857  cbviun  3858  iunss  3862  uniiun  3874  iunopab  4211  opeliunxp  4602  reliun  4668  fnasrn  5606  fnasrng  5608  abrexex2g  6026  abrexex2  6030  bdciun  13247