Theorem bdciun 15776

Description: The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdciun.1	|- BOUNDED A

Assertion

Ref	Expression
bdciun	|- BOUNDED U_ x e. y A

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem bdciun

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdciun.1	. . . . 5 |- BOUNDED A
2	1	bdeli 15744	. . . 4 |- BOUNDED z e. A
3	2	ax-bdex 15717	. . 3 |- BOUNDED E. x e. y z e. A
4	3	bdcab 15747	. 2 |- BOUNDED { z | E. x e. y z e. A }
5		df-iun 3928	. 2 |- U_ x e. y A = { z | E. x e. y z e. A }
6	4, 5	bdceqir 15742	1 |- BOUNDED U_ x e. y A

Syntax hints:   e. wcel 2175   {cab 2190   E.wrex 2484   U_ciun 3926  BOUNDED wbdc 15738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186  ax-bd0 15711  ax-bdex 15717  ax-bdsb 15720

This theorem depends on definitions:  df-bi 117  df-clab 2191  df-cleq 2197  df-clel 2200  df-iun 3928  df-bdc 15739

This theorem is referenced by: (None)