Theorem bdciin 16014

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 15981	. . . 4 |- BOUNDED z e. A
3	2	ax-bdal 15953	. . 3 |- BOUNDED A. x e. y z e. A
4	3	bdcab 15984	. 2 |- BOUNDED { z | A. x e. y z e. A }
5		df-iin 3944	. 2 |- |^|_ x e. y A = { z | A. x e. y z e. A }
6	4, 5	bdceqir 15979	1 |- BOUNDED |^|_ x e. y A

Syntax hints:   e. wcel 2178   {cab 2193   A.wral 2486   |^|_ciin 3942  BOUNDED wbdc 15975

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558  ax-ext 2189  ax-bd0 15948  ax-bdal 15953  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-clab 2194  df-cleq 2200  df-clel 2203  df-iin 3944  df-bdc 15976

This theorem is referenced by: (None)