Theorem bdcint 16634

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 16578	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdal 16575	. . . 4 |- BOUNDED A. z e. x y e. z
3		df-ral 2525	. . . 4 |- ( A. z e. x y e. z <-> A. z ( z e. x -> y e. z ) )
4	2, 3	bd0 16581	. . 3 |- BOUNDED A. z ( z e. x -> y e. z )
5	4	bdcab 16606	. 2 |- BOUNDED { y | A. z ( z e. x -> y e. z ) }
6		df-int 3949	. 2 |- |^| x = { y | A. z ( z e. x -> y e. z ) }
7	5, 6	bdceqir 16601	1 |- BOUNDED |^| x

Syntax hints:   -> wi 4   A.wal 1396   {cab 2218   A.wral 2520   |^|cint 3948  BOUNDED wbdc 16597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214  ax-bd0 16570  ax-bdal 16575  ax-bdel 16578  ax-bdsb 16579

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-cleq 2225  df-clel 2228  df-ral 2525  df-int 3949  df-bdc 16598

This theorem is referenced by: (None)