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Theorem bdcrab 15042
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 {𝑥𝐴𝜑}
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem bdcrab
StepHypRef Expression
1 bdcrab.1 . . . . 5 BOUNDED 𝐴
21bdeli 15036 . . . 4 BOUNDED 𝑥𝐴
3 bdcrab.2 . . . 4 BOUNDED 𝜑
42, 3ax-bdan 15005 . . 3 BOUNDED (𝑥𝐴𝜑)
54bdcab 15039 . 2 BOUNDED {𝑥 ∣ (𝑥𝐴𝜑)}
6 df-rab 2477 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
75, 6bdceqir 15034 1 BOUNDED {𝑥𝐴𝜑}
Colors of variables: wff set class
Syntax hints:  wa 104  wcel 2160  {cab 2175  {crab 2472  BOUNDED wbd 15002  BOUNDED wbdc 15030
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2171  ax-bd0 15003  ax-bdan 15005  ax-bdsb 15012
This theorem depends on definitions:  df-bi 117  df-clab 2176  df-cleq 2182  df-clel 2185  df-rab 2477  df-bdc 15031
This theorem is referenced by:  bdrabexg  15096
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