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Theorem bdcrab 15090
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  ph
Assertion
Ref Expression
bdcrab  |- BOUNDED  { x  e.  A  |  ph }
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem bdcrab
StepHypRef Expression
1 bdcrab.1 . . . . 5  |- BOUNDED  A
21bdeli 15084 . . . 4  |- BOUNDED  x  e.  A
3 bdcrab.2 . . . 4  |- BOUNDED  ph
42, 3ax-bdan 15053 . . 3  |- BOUNDED  ( x  e.  A  /\  ph )
54bdcab 15087 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  ph ) }
6 df-rab 2477 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
75, 6bdceqir 15082 1  |- BOUNDED  { x  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 104    e. wcel 2160   {cab 2175   {crab 2472  BOUNDED wbd 15050  BOUNDED wbdc 15078
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 15051  ax-bdan 15053  ax-bdsb 15060
This theorem depends on definitions:  df-bi 117  df-clab 2176  df-cleq 2182  df-clel 2185  df-rab 2477  df-bdc 15079
This theorem is referenced by:  bdrabexg  15144
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