Theorem bdcint 14714

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 14658	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdal 14655	. . . 4 |- BOUNDED A. z e. x y e. z
3		df-ral 2460	. . . 4 |- ( A. z e. x y e. z <-> A. z ( z e. x -> y e. z ) )
4	2, 3	bd0 14661	. . 3 |- BOUNDED A. z ( z e. x -> y e. z )
5	4	bdcab 14686	. 2 |- BOUNDED { y | A. z ( z e. x -> y e. z ) }
6		df-int 3847	. 2 |- |^| x = { y | A. z ( z e. x -> y e. z ) }
7	5, 6	bdceqir 14681	1 |- BOUNDED |^| x

Syntax hints:   -> wi 4   A.wal 1351   {cab 2163   A.wral 2455   |^|cint 3846  BOUNDED wbdc 14677

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14650  ax-bdal 14655  ax-bdel 14658  ax-bdsb 14659

This theorem depends on definitions:  df-bi 117  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-int 3847  df-bdc 14678

This theorem is referenced by: (None)