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Theorem bdcrab 10801
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 10795 . . . 4 BOUNDED 𝑥𝐴
3 bdcrab.2 . . . 4 BOUNDED 𝜑
42, 3ax-bdan 10764 . . 3 BOUNDED (𝑥𝐴𝜑)
54bdcab 10798 . 2 BOUNDED {𝑥 ∣ (𝑥𝐴𝜑)}
6 df-rab 2358 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
75, 6bdceqir 10793 1 BOUNDED {𝑥𝐴𝜑}
Colors of variables: wff set class
Syntax hints:  wa 102  wcel 1434  {cab 2068  {crab 2353  BOUNDED wbd 10761  BOUNDED wbdc 10789
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064  ax-bd0 10762  ax-bdan 10764  ax-bdsb 10771
This theorem depends on definitions:  df-bi 115  df-clab 2069  df-cleq 2075  df-clel 2078  df-rab 2358  df-bdc 10790
This theorem is referenced by:  bdrabexg  10855
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