Theorem bdciin 13771

Description: The indexed intersection 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
bdciin	|- BOUNDED |^|_ x e. y A

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem bdciin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdciun.1	. . . . 5 |- BOUNDED A
2	1	bdeli 13738	. . . 4 |- BOUNDED z e. A
3	2	ax-bdal 13710	. . 3 |- BOUNDED A. x e. y z e. A
4	3	bdcab 13741	. 2 |- BOUNDED { z | A. x e. y z e. A }
5		df-iin 3869	. 2 |- |^|_ x e. y A = { z | A. x e. y z e. A }
6	4, 5	bdceqir 13736	1 |- BOUNDED |^|_ x e. y A

Syntax hints:   e. wcel 2136   {cab 2151   A.wral 2444   |^|_ciin 3867  BOUNDED wbdc 13732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13705  ax-bdal 13710  ax-bdsb 13714

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-cleq 2158  df-clel 2161  df-iin 3869  df-bdc 13733

This theorem is referenced by: (None)