Theorem bj-bdcel 16130

Description: Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)

Hypothesis

Ref	Expression
bj-bdcel.bd	|- BOUNDED y = A

Assertion

Ref	Expression
bj-bdcel	|- BOUNDED A e. x

Distinct variable groups:   x, y   y, A

Allowed substitution hint:   A( x)

Proof of Theorem bj-bdcel

Step	Hyp	Ref	Expression
1		bj-bdcel.bd	. . 3 |- BOUNDED y = A
2	1	ax-bdex 16112	. 2 |- BOUNDED E. y e. x y = A
3		risset 2558	. 2 |- ( A e. x <-> E. y e. x y = A )
4	2, 3	bd0r 16118	1 |- BOUNDED A e. x

Syntax hints:   = wceq 1395   e. wcel 2200   E.wrex 2509  BOUNDED wbd 16105

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580  ax-bd0 16106  ax-bdex 16112

This theorem depends on definitions:  df-bi 117  df-clel 2225  df-rex 2514

This theorem is referenced by:  bj-bd0el  16161  bj-bdsucel  16175