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Theorem opabbrex 5631
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  e.  _V  /\  E  e.  _V )  ->  ( f ( V W E ) p  ->  th ) )
opabbrex.2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  th }  e.  _V )
Assertion
Ref Expression
opabbrex  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V W E ) p  /\  ps ) }  e.  _V )
Distinct variable groups:    f, E, p   
f, V, p
Allowed substitution hints:    ps( f, p)    th( f, p)    W( f, p)

Proof of Theorem opabbrex
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-opab 3869 . . 3  |-  { <. f ,  p >.  |  th }  =  { z  |  E. f E. p
( z  =  <. f ,  p >.  /\  th ) }
2 opabbrex.2 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  th }  e.  _V )
31, 2syl5eqelr 2172 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { z  |  E. f E. p ( z  =  <. f ,  p >.  /\  th ) }  e.  _V )
4 df-opab 3869 . . 3  |-  { <. f ,  p >.  |  ( f ( V W E ) p  /\  ps ) }  =  {
z  |  E. f E. p ( z  = 
<. f ,  p >.  /\  ( f ( V W E ) p  /\  ps ) ) }
5 opabbrex.1 . . . . . . 7  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( f ( V W E ) p  ->  th ) )
65adantrd 273 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( f ( V W E ) p  /\  ps )  ->  th ) )
76anim2d 330 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( z  = 
<. f ,  p >.  /\  ( f ( V W E ) p  /\  ps ) )  ->  ( z  = 
<. f ,  p >.  /\ 
th ) ) )
872eximdv 1807 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( E. f E. p ( z  = 
<. f ,  p >.  /\  ( f ( V W E ) p  /\  ps ) )  ->  E. f E. p
( z  =  <. f ,  p >.  /\  th ) ) )
98ss2abdv 3080 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { z  |  E. f E. p ( z  =  <. f ,  p >.  /\  ( f ( V W E ) p  /\  ps )
) }  C_  { z  |  E. f E. p ( z  = 
<. f ,  p >.  /\ 
th ) } )
104, 9syl5eqss 3056 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V W E ) p  /\  ps ) }  C_  { z  |  E. f E. p
( z  =  <. f ,  p >.  /\  th ) } )
113, 10ssexd 3947 1  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { <. f ,  p >.  |  ( f ( V W E ) p  /\  ps ) }  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287   E.wex 1424    e. wcel 1436   {cab 2071   _Vcvv 2614   <.cop 3428   class class class wbr 3814   {copab 3867  (class class class)co 5594
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-in 2992  df-ss 2999  df-opab 3869
This theorem is referenced by:  sprmpt2  5942
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