Theorem bdcrab 12729

Description: A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdcrab.1	⊢ BOUNDED 𝐴
bdcrab.2	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdcrab	⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bdcrab

Step	Hyp	Ref	Expression
1		bdcrab.1	. . . . 5 ⊢ BOUNDED 𝐴
2	1	bdeli 12723	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcrab.2	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	ax-bdan 12692	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝜑)
5	4	bdcab 12726	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6		df-rab 2397	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	5, 6	bdceqir 12721	1 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 103   ∈ wcel 1461  {cab 2099  {crab 2392  BOUNDED wbd 12689  BOUNDED wbdc 12717

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-17 1487  ax-ial 1495  ax-ext 2095  ax-bd0 12690  ax-bdan 12692  ax-bdsb 12699

This theorem depends on definitions:  df-bi 116  df-clab 2100  df-cleq 2106  df-clel 2109  df-rab 2397  df-bdc 12718

This theorem is referenced by:  bdrabexg  12783