Theorem bdciin 16474

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 16441	. . . 4 |- BOUNDED z e. A
3	2	ax-bdal 16413	. . 3 |- BOUNDED A. x e. y z e. A
4	3	bdcab 16444	. 2 |- BOUNDED { z | A. x e. y z e. A }
5		df-iin 3973	. 2 |- |^|_ x e. y A = { z | A. x e. y z e. A }
6	4, 5	bdceqir 16439	1 |- BOUNDED |^|_ x e. y A

Syntax hints:   e. wcel 2202   {cab 2217   A.wral 2510   |^|_ciin 3971  BOUNDED wbdc 16435

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213  ax-bd0 16408  ax-bdal 16413  ax-bdsb 16417

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-cleq 2224  df-clel 2227  df-iin 3973  df-bdc 16436

This theorem is referenced by: (None)