Theorem bdciun 11769

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 11737	. . . 4 |- BOUNDED z e. A
3	2	ax-bdex 11710	. . 3 |- BOUNDED E. x e. y z e. A
4	3	bdcab 11740	. 2 |- BOUNDED { z | E. x e. y z e. A }
5		df-iun 3732	. 2 |- U_ x e. y A = { z | E. x e. y z e. A }
6	4, 5	bdceqir 11735	1 |- BOUNDED U_ x e. y A

Syntax hints:   e. wcel 1438   {cab 2074   E.wrex 2360   U_ciun 3730  BOUNDED wbdc 11731

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11704  ax-bdex 11710  ax-bdsb 11713

This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-iun 3732  df-bdc 11732

This theorem is referenced by: (None)