Theorem bdciin 10670

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 10637	. . . 4 |- BOUNDED z e. A
3	2	ax-bdal 10609	. . 3 |- BOUNDED A. x e. y z e. A
4	3	bdcab 10640	. 2 |- BOUNDED { z | A. x e. y z e. A }
5		df-iin 3681	. 2 |- |^|_ x e. y A = { z | A. x e. y z e. A }
6	4, 5	bdceqir 10635	1 |- BOUNDED |^|_ x e. y A

Syntax hints:   e. wcel 1433   {cab 2067   A.wral 2348   |^|_ciin 3679  BOUNDED wbdc 10631

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 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-bd0 10604  ax-bdal 10609  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-iin 3681  df-bdc 10632

This theorem is referenced by: (None)