Definition df-iun 3915

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 3947. Theorem uniiun 3967 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 3913	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1363	. . . . 5 class y
7	6, 3	wcel 2164	. . . 4 wff y e. B
8	7, 1, 2	wrex 2473	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2179	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1364	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3917  nfiunxy  3939  nfiunya  3941  nfiu1  3943  dfiunv2  3949  cbviun  3950  iunss  3954  uniiun  3967  iunopab  4313  opeliunxp  4715  reliun  4781  fnasrn  5737  fnasrng  5739  abrexex2g  6174  abrexex2  6178  bdciun  15440