Theorem bj-bdcel 10628

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 ∈ 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 10610	. 2 ⊢ BOUNDED ∃𝑦 ∈ 𝑥 𝑦 = 𝐴
3		risset 2394	. 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝑦 = 𝐴)
4	2, 3	bd0r 10616	1 ⊢ BOUNDED 𝐴 ∈ 𝑥

Syntax hints:   = wceq 1284   ∈ wcel 1433  ∃wrex 2349  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467  ax-bd0 10604  ax-bdex 10610

This theorem depends on definitions:  df-bi 115  df-clel 2077  df-rex 2354

This theorem is referenced by:  bj-bd0el  10659  bj-bdsucel  10673