Theorem bdciin 13914

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 13881	. . . 4 |- BOUNDED z e. A
3	2	ax-bdal 13853	. . 3 |- BOUNDED A. x e. y z e. A
4	3	bdcab 13884	. 2 |- BOUNDED { z | A. x e. y z e. A }
5		df-iin 3876	. 2 |- |^|_ x e. y A = { z | A. x e. y z e. A }
6	4, 5	bdceqir 13879	1 |- BOUNDED |^|_ x e. y A

Syntax hints:   e. wcel 2141   {cab 2156   A.wral 2448   |^|_ciin 3874  BOUNDED wbdc 13875

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152  ax-bd0 13848  ax-bdal 13853  ax-bdsb 13857

This theorem depends on definitions:  df-bi 116  df-clab 2157  df-cleq 2163  df-clel 2166  df-iin 3876  df-bdc 13876

This theorem is referenced by: (None)