Theorem bdcrab 11400

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 11394	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcrab.2	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	ax-bdan 11363	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝜑)
5	4	bdcab 11397	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6		df-rab 2368	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	5, 6	bdceqir 11392	1 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 102   ∈ wcel 1438  {cab 2074  {crab 2363  BOUNDED wbd 11360  BOUNDED wbdc 11388

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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11361  ax-bdan 11363  ax-bdsb 11370

This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-rab 2368  df-bdc 11389

This theorem is referenced by:  bdrabexg  11454