Theorem bj-bdcel 9957

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 9939	. 2 |- BOUNDED E. y e. x y = A
3		risset 2352	. 2 |- ( A e. x <-> E. y e. x y = A )
4	2, 3	bd0r 9945	1 |- BOUNDED A e. x

Syntax hints:   = wceq 1243   e. wcel 1393   E.wrex 2307  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-bd0 9933  ax-bdex 9939

This theorem depends on definitions:  df-bi 110  df-clel 2036  df-rex 2312

This theorem is referenced by:  bj-bd0el  9988  bj-bdsucel  10002