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Theorem bdrabexg 13031
Description: Bounded version of rabexg 4041. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdrabexg.bd  |- BOUNDED  ph
bdrabexg.bdc  |- BOUNDED  A
Assertion
Ref Expression
bdrabexg  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem bdrabexg
StepHypRef Expression
1 ssrab2 3152 . 2  |-  { x  e.  A  |  ph }  C_  A
2 bdrabexg.bdc . . . 4  |- BOUNDED  A
3 bdrabexg.bd . . . 4  |- BOUNDED  ph
42, 3bdcrab 12977 . . 3  |- BOUNDED  { x  e.  A  |  ph }
54bdssexg 13029 . 2  |-  ( ( { x  e.  A  |  ph }  C_  A  /\  A  e.  V
)  ->  { x  e.  A  |  ph }  e.  _V )
61, 5mpan 420 1  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465   {crab 2397   _Vcvv 2660    C_ wss 3041  BOUNDED wbd 12937  BOUNDED wbdc 12965
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bd0 12938  ax-bdan 12940  ax-bdsb 12947  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rab 2402  df-v 2662  df-in 3047  df-ss 3054  df-bdc 12966
This theorem is referenced by:  bj-inex  13032
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