Definition df-iun 3914

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 3946. Theorem uniiun 3966 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 3912	. 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  3916  nfiunxy  3938  nfiunya  3940  nfiu1  3942  dfiunv2  3948  cbviun  3949  iunss  3953  uniiun  3966  iunopab  4312  opeliunxp  4714  reliun  4780  fnasrn  5736  fnasrng  5738  abrexex2g  6172  abrexex2  6176  bdciun  15370