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Theorem bdcrab 11188
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 11182 . . . 4  |- BOUNDED  x  e.  A
3 bdcrab.2 . . . 4  |- BOUNDED  ph
42, 3ax-bdan 11151 . . 3  |- BOUNDED  ( x  e.  A  /\  ph )
54bdcab 11185 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  ph ) }
6 df-rab 2364 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
75, 6bdceqir 11180 1  |- BOUNDED  { x  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1436   {cab 2071   {crab 2359  BOUNDED wbd 11148  BOUNDED wbdc 11176
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-17 1462  ax-ial 1470  ax-ext 2067  ax-bd0 11149  ax-bdan 11151  ax-bdsb 11158
This theorem depends on definitions:  df-bi 115  df-clab 2072  df-cleq 2078  df-clel 2081  df-rab 2364  df-bdc 11177
This theorem is referenced by:  bdrabexg  11242
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