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Theorem bdrabexg 11680
Description: Bounded version of rabexg 3980. (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 3106 . 2  |-  { x  e.  A  |  ph }  C_  A
2 bdrabexg.bdc . . . 4  |- BOUNDED  A
3 bdrabexg.bd . . . 4  |- BOUNDED  ph
42, 3bdcrab 11626 . . 3  |- BOUNDED  { x  e.  A  |  ph }
54bdssexg 11678 . 2  |-  ( ( { x  e.  A  |  ph }  C_  A  /\  A  e.  V
)  ->  { x  e.  A  |  ph }  e.  _V )
61, 5mpan 415 1  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1438   {crab 2363   _Vcvv 2619    C_ wss 2999  BOUNDED wbd 11586  BOUNDED wbdc 11614
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11587  ax-bdan 11589  ax-bdsb 11596  ax-bdsep 11658
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rab 2368  df-v 2621  df-in 3005  df-ss 3012  df-bdc 11615
This theorem is referenced by:  bj-inex  11681
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