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

Proof of Theorem bdcrab
StepHypRef Expression
1 bdcrab.1 . . . . 5 BOUNDED 𝐴
21bdeli 9966 . . . 4 BOUNDED 𝑥𝐴
3 bdcrab.2 . . . 4 BOUNDED 𝜑
42, 3ax-bdan 9935 . . 3 BOUNDED (𝑥𝐴𝜑)
54bdcab 9969 . 2 BOUNDED {𝑥 ∣ (𝑥𝐴𝜑)}
6 df-rab 2315 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
75, 6bdceqir 9964 1 BOUNDED {𝑥𝐴𝜑}
 Colors of variables: wff set class 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
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