Theorem bdcint 12041

Description: The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)

Assertion

Ref	Expression
bdcint	|- BOUNDED |^| x

Proof of Theorem bdcint

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-bdel 11985	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdal 11982	. . . 4 |- BOUNDED A. z e. x y e. z
3		df-ral 2365	. . . 4 |- ( A. z e. x y e. z <-> A. z ( z e. x -> y e. z ) )
4	2, 3	bd0 11988	. . 3 |- BOUNDED A. z ( z e. x -> y e. z )
5	4	bdcab 12013	. 2 |- BOUNDED { y | A. z ( z e. x -> y e. z ) }
6		df-int 3695	. 2 |- |^| x = { y | A. z ( z e. x -> y e. z ) }
7	5, 6	bdceqir 12008	1 |- BOUNDED |^| x

Syntax hints:   -> wi 4   A.wal 1288   {cab 2075   A.wral 2360   |^|cint 3694  BOUNDED wbdc 12004

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-ext 2071  ax-bd0 11977  ax-bdal 11982  ax-bdel 11985  ax-bdsb 11986

This theorem depends on definitions:  df-bi 116  df-clab 2076  df-cleq 2082  df-clel 2085  df-ral 2365  df-int 3695  df-bdc 12005

This theorem is referenced by: (None)