Definition df-iun 3906

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 3938. Theorem uniiun 3958 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 3904	. 2 class U_ x e. A B
5		vy	. . . . . 6 setvar y
6	5	cv 1363	. . . . 5 class y
7	6, 3	wcel 2160	. . . 4 wff y e. B
8	7, 1, 2	wrex 2469	. . 3 wff E. x e. A y e. B
9	8, 5	cab 2175	. 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  3908  nfiunxy  3930  nfiunya  3932  nfiu1  3934  dfiunv2  3940  cbviun  3941  iunss  3945  uniiun  3958  iunopab  4302  opeliunxp  4702  reliun  4768  fnasrn  5718  fnasrng  5720  abrexex2g  6149  abrexex2  6153  bdciun  15116