Theorem bdcrab 13039

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 13033	. . . 4 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		bdcrab.2	. . . 4 ⊢ BOUNDED 𝜑
4	2, 3	ax-bdan 13002	. . 3 ⊢ BOUNDED (𝑥 ∈ 𝐴 ∧ 𝜑)
5	4	bdcab 13036	. 2 ⊢ BOUNDED {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
6		df-rab 2423	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
7	5, 6	bdceqir 13031	1 ⊢ BOUNDED {𝑥 ∈ 𝐴 ∣ 𝜑}

Syntax hints:   ∧ wa 103   ∈ wcel 1480  {cab 2123  {crab 2418  BOUNDED wbd 12999  BOUNDED wbdc 13027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119  ax-bd0 13000  ax-bdan 13002  ax-bdsb 13009

This theorem depends on definitions:  df-bi 116  df-clab 2124  df-cleq 2130  df-clel 2133  df-rab 2423  df-bdc 13028

This theorem is referenced by:  bdrabexg  13093