Theorem bj-bdcel 16961

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 16943	. 2 ⊢ BOUNDED ∃𝑦 ∈ 𝑥 𝑦 = 𝐴
3		risset 2578	. 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝑦 = 𝐴)
4	2, 3	bd0r 16949	1 ⊢ BOUNDED 𝐴 ∈ 𝑥

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  ∃wrex 2529  BOUNDED wbd 16936

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587  ax-bd0 16937  ax-bdex 16943

This proof depends on definitions:  df-bi 117  df-clel 2234  df-rex 2534

This theorem is used by:  bj-bd0el  16992  bj-bdsucel  17006