Theorem bdcint 10668

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 10612	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdal 10609	. . . 4 |- BOUNDED A. z e. x y e. z
3		df-ral 2353	. . . 4 |- ( A. z e. x y e. z <-> A. z ( z e. x -> y e. z ) )
4	2, 3	bd0 10615	. . 3 |- BOUNDED A. z ( z e. x -> y e. z )
5	4	bdcab 10640	. 2 |- BOUNDED { y | A. z ( z e. x -> y e. z ) }
6		df-int 3637	. 2 |- |^| x = { y | A. z ( z e. x -> y e. z ) }
7	5, 6	bdceqir 10635	1 |- BOUNDED |^| x

Syntax hints:   -> wi 4   A.wal 1282   {cab 2067   A.wral 2348   |^|cint 3636  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-bdel 10612  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-int 3637  df-bdc 10632

This theorem is referenced by: (None)