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Theorem opabbrex 5491
Description: A collection of ordered pairs with an extension of a binary relation is a set. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
Hypotheses
Ref Expression
opabbrex.1  V  _V  E  _V  V W E p
opabbrex.2  V  _V  E  _V  { <. ,  p >.  |  }  _V
Assertion
Ref Expression
opabbrex  V  _V  E  _V  { <. ,  p >.  |  V W E p  }  _V
Distinct variable groups:   , E, p   , V, p
Allowed substitution hints:   (, p)   (, p)    W(, p)

Proof of Theorem opabbrex
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-opab 3810 . . 3  { <. ,  p >.  |  }  {  |  p  <. ,  p >.  }
2 opabbrex.2 . . 3  V  _V  E  _V  { <. ,  p >.  |  }  _V
31, 2syl5eqelr 2122 . 2  V  _V  E  _V  {  |  p  <. ,  p >.  }  _V
4 df-opab 3810 . . 3  { <. ,  p >.  |  V W E p  }  {  |  p 
<. ,  p >.  V W E p  }
5 opabbrex.1 . . . . . . 7  V  _V  E  _V  V W E p
65adantrd 264 . . . . . 6  V  _V  E  _V  V W E p
76anim2d 320 . . . . 5  V  _V  E  _V 
<. ,  p >.  V W E p  <. ,  p >.
872eximdv 1759 . . . 4  V  _V  E  _V  p 
<. ,  p >.  V W E p  p  <. ,  p >.
98ss2abdv 3007 . . 3  V  _V  E  _V  {  |  p  <. ,  p >.  V W E p  }  C_  {  |  p 
<. ,  p >.  }
104, 9syl5eqss 2983 . 2  V  _V  E  _V  { <. ,  p >.  |  V W E p  }  C_  {  |  p  <. ,  p >.  }
113, 10ssexd 3888 1  V  _V  E  _V  { <. ,  p >.  |  V W E p  }  _V
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   {cab 2023   _Vcvv 2551   <.cop 3370   class class class wbr 3755   {copab 3808  (class class class)co 5455
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-opab 3810
This theorem is referenced by:  sprmpt2  5798
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