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Theorem bdrabexg 10026
Description: Bounded version of rabexg 3900. (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 3025 . 2  |-  { x  e.  A  |  ph }  C_  A
2 bdrabexg.bdc . . . 4  |- BOUNDED  A
3 bdrabexg.bd . . . 4  |- BOUNDED  ph
42, 3bdcrab 9972 . . 3  |- BOUNDED  { x  e.  A  |  ph }
54bdssexg 10024 . 2  |-  ( ( { x  e.  A  |  ph }  C_  A  /\  A  e.  V
)  ->  { x  e.  A  |  ph }  e.  _V )
61, 5mpan 400 1  |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1393   {crab 2310   _Vcvv 2557    C_ wss 2917  BOUNDED wbd 9932  BOUNDED wbdc 9960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdsb 9942  ax-bdsep 10004
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-bdc 9961
This theorem is referenced by:  bj-inex  10027
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