Theorem bdcint 15369

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 15313	. . . . 5 |- BOUNDED y e. z
2	1	ax-bdal 15310	. . . 4 |- BOUNDED A. z e. x y e. z
3		df-ral 2477	. . . 4 |- ( A. z e. x y e. z <-> A. z ( z e. x -> y e. z ) )
4	2, 3	bd0 15316	. . 3 |- BOUNDED A. z ( z e. x -> y e. z )
5	4	bdcab 15341	. 2 |- BOUNDED { y | A. z ( z e. x -> y e. z ) }
6		df-int 3871	. 2 |- |^| x = { y | A. z ( z e. x -> y e. z ) }
7	5, 6	bdceqir 15336	1 |- BOUNDED |^| x

Syntax hints:   -> wi 4   A.wal 1362   {cab 2179   A.wral 2472   |^|cint 3870  BOUNDED wbdc 15332

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175  ax-bd0 15305  ax-bdal 15310  ax-bdel 15313  ax-bdsb 15314

This theorem depends on definitions:  df-bi 117  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-int 3871  df-bdc 15333

This theorem is referenced by: (None)