Definition df-iun 2563

Description: Define indexed union. Definition of [Stoll] p. 45. In normal use, A is independent of x, 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 distinct variable group (meaning A cannot depend on x) and that B and x do not share a distinct 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 2582. Theorem uniiun 2596 provides a definition of ordinary union in terms of indexed union. Theorems fniunfv 3856 and funiunfv 3857 are useful when B is a function.

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

Detailed syntax breakdown of Definition df-iun

Step	Hyp	Ref	Expression
1		vx	. . 3 set x
2		cA	. . 3 class A
3		cB	. . 3 class B
4	1, 2, 3	ciun 2561	. 2 class U_x e. A B
5		vy	. . . . . 6 set y
6	5	cv 953	. . . . 5 class y
7	6, 3	wcel 956	. . . 4 wff y e. B
8	7, 1, 2	wrex 1643	. . 3 wff E.x e. A y e. B
9	8, 5	cab 1461	. 2 class {y | E.x e. A y e. B}
10	4, 9	wceq 954	1 wff U_x e. A B = {y | E.x e. A y e. B}

This definition is referenced by:  eliun 2565  iunss1 2569  hbiu1 2579  dfiun2g 2581  cbviun 2584  uniiun 2596  iunun 2608  abrexex2 3862

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