Theorem bj-bdcel 16432

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 16414	. 2 ⊢ BOUNDED ∃𝑦 ∈ 𝑥 𝑦 = 𝐴
3		risset 2560	. 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝑦 = 𝐴)
4	2, 3	bd0r 16420	1 ⊢ BOUNDED 𝐴 ∈ 𝑥

Syntax hints:   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  BOUNDED wbd 16407

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 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-ial 1582  ax-bd0 16408  ax-bdex 16414

This theorem depends on definitions:  df-bi 117  df-clel 2227  df-rex 2516

This theorem is referenced by:  bj-bd0el  16463  bj-bdsucel  16477