Theorem bdcrab 16507

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 16501	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcrab.2	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	ax-bdan 16470	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝜑)
5	4	bdcab 16504	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6		df-rab 2518	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	5, 6	bdceqir 16499	1 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 104   ∈ wcel 2201  {cab 2216  {crab 2513  BOUNDED wbd 16467  BOUNDED wbdc 16495

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-17 1574  ax-ial 1582  ax-ext 2212  ax-bd0 16468  ax-bdan 16470  ax-bdsb 16477

This theorem depends on definitions:  df-bi 117  df-clab 2217  df-cleq 2223  df-clel 2226  df-rab 2518  df-bdc 16496

This theorem is referenced by:  bdrabexg  16561