Theorem bdcrab 9972

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 9966	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcrab.2	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	ax-bdan 9935	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝜑)
5	4	bdcab 9969	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6		df-rab 2315	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	5, 6	bdceqir 9964	1 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 97   ∈ wcel 1393  {cab 2026  {crab 2310  BOUNDED wbd 9932  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-rab 2315  df-bdc 9961

This theorem is referenced by:  bdrabexg  10026