Definition df-iun 3943

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 3975. Theorem uniiun 3995 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 3941	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1372	. . . . 5 class y
7	6, 3	wcel 2178	. . . 4 wff y e. B
8	7, 1, 2	wrex 2487	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2193	. 2 class { y | E. x e. A y e. B }
10	4, 9	wceq 1373	1 wff U_ x e. A B = { y | E. x e. A y e. B }

This definition is referenced by:  eliun  3945  nfiunxy  3967  nfiunya  3969  nfiu1  3971  dfiunv2  3977  cbviun  3978  iunss  3982  uniiun  3995  iunopab  4346  opeliunxp  4748  reliun  4814  fnasrn  5781  fnasrng  5783  abrexex2g  6228  abrexex2  6232  bdciun  16013