Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdrabexg Unicode version

Theorem bdrabexg 11140
Description: Bounded version of rabexg 3947. (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 3090 . 2  |-  { x  e.  A  |  ph }  C_  A
2 bdrabexg.bdc . . . 4  |- BOUNDED  A
3 bdrabexg.bd . . . 4  |- BOUNDED  ph
42, 3bdcrab 11086 . . 3  |- BOUNDED  { x  e.  A  |  ph }
54bdssexg 11138 . 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 1434   {crab 2357   _Vcvv 2612    C_ wss 2984  BOUNDED wbd 11046  BOUNDED wbdc 11074
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-bd0 11047  ax-bdan 11049  ax-bdsb 11056  ax-bdsep 11118
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rab 2362  df-v 2614  df-in 2990  df-ss 2997  df-bdc 11075
This theorem is referenced by:  bj-inex  11141
  Copyright terms: Public domain W3C validator