Theorem bdcint 13769

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 13713	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdal 13710	. . . 4 |- BOUNDED A. z e. x y e. z
3		df-ral 2449	. . . 4 |- ( A. z e. x y e. z <-> A. z ( z e. x -> y e. z ) )
4	2, 3	bd0 13716	. . 3 |- BOUNDED A. z ( z e. x -> y e. z )
5	4	bdcab 13741	. 2 |- BOUNDED { y | A. z ( z e. x -> y e. z ) }
6		df-int 3825	. 2 |- |^| x = { y | A. z ( z e. x -> y e. z ) }
7	5, 6	bdceqir 13736	1 |- BOUNDED |^| x

Syntax hints:   -> wi 4   A.wal 1341   {cab 2151   A.wral 2444   |^|cint 3824  BOUNDED wbdc 13732

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13705  ax-bdal 13710  ax-bdel 13713  ax-bdsb 13714

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-int 3825  df-bdc 13733

This theorem is referenced by: (None)