Theorem bdcint 16915

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 16859	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdal 16856	. . . 4 |- BOUNDED A. z e. x y e. z
3		df-ral 2533	. . . 4 |- ( A. z e. x y e. z <-> A. z ( z e. x -> y e. z ) )
4	2, 3	bd0 16862	. . 3 |- BOUNDED A. z ( z e. x -> y e. z )
5	4	bdcab 16887	. 2 |- BOUNDED { y | A. z ( z e. x -> y e. z ) }
6		df-int 3971	. 2 |- |^| x = { y | A. z ( z e. x -> y e. z ) }
7	5, 6	bdceqir 16882	1 |- BOUNDED |^| x

This proof depends on syntax axioms:   -> wi 4   A.wal 1400   {cab 2224   A.wral 2528   |^|cint 3970  BOUNDED wbdc 16878

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16851  ax-bdal 16856  ax-bdel 16859  ax-bdsb 16860

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-int 3971  df-bdc 16879

This theorem is used by: (None)