Theorem bj-bdcel 11166

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 11148	. 2 ⊢ BOUNDED ∃𝑦 ∈ 𝑥 𝑦 = 𝐴
3		risset 2402	. 2 ⊢ (𝐴 ∈ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝑦 = 𝐴)
4	2, 3	bd0r 11154	1 ⊢ BOUNDED 𝐴 ∈ 𝑥

Syntax hints:   = wceq 1287   ∈ wcel 1436  ∃wrex 2356  BOUNDED wbd 11141

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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470  ax-bd0 11142  ax-bdex 11148

This theorem depends on definitions:  df-bi 115  df-clel 2081  df-rex 2361

This theorem is referenced by:  bj-bd0el  11197  bj-bdsucel  11211