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Theorem bdcrab 13155
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 13149 . . . 4  |- BOUNDED  x  e.  A
3 bdcrab.2 . . . 4  |- BOUNDED  ph
42, 3ax-bdan 13118 . . 3  |- BOUNDED  ( x  e.  A  /\  ph )
54bdcab 13152 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  ph ) }
6 df-rab 2425 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
75, 6bdceqir 13147 1  |- BOUNDED  { x  e.  A  |  ph }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 1480   {cab 2125   {crab 2420  BOUNDED wbd 13115  BOUNDED wbdc 13143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121  ax-bd0 13116  ax-bdan 13118  ax-bdsb 13125
This theorem depends on definitions:  df-bi 116  df-clab 2126  df-cleq 2132  df-clel 2135  df-rab 2425  df-bdc 13144
This theorem is referenced by:  bdrabexg  13209
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