Theorem bdcint 13246

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 13190	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdal 13187	. . . 4 |- BOUNDED A. z e. x y e. z
3		df-ral 2422	. . . 4 |- ( A. z e. x y e. z <-> A. z ( z e. x -> y e. z ) )
4	2, 3	bd0 13193	. . 3 |- BOUNDED A. z ( z e. x -> y e. z )
5	4	bdcab 13218	. 2 |- BOUNDED { y | A. z ( z e. x -> y e. z ) }
6		df-int 3780	. 2 |- |^| x = { y | A. z ( z e. x -> y e. z ) }
7	5, 6	bdceqir 13213	1 |- BOUNDED |^| x

Syntax hints:   -> wi 4   A.wal 1330   {cab 2126   A.wral 2417   |^|cint 3779  BOUNDED wbdc 13209

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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-ial 1515  ax-ext 2122  ax-bd0 13182  ax-bdal 13187  ax-bdel 13190  ax-bdsb 13191

This theorem depends on definitions:  df-bi 116  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-int 3780  df-bdc 13210

This theorem is referenced by: (None)