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Theorem opabbrex 5912
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 4062 . . 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, 2eqeltrrid 2265 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  { z  |  E. f E. p ( z  =  <. f ,  p >.  /\  th ) }  e.  _V )
4 df-opab 4062 . . 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 279 . . . . . 6  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( f ( V W E ) p  /\  ps )  ->  th ) )
76anim2d 337 . . . . 5  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( z  = 
<. f ,  p >.  /\  ( f ( V W E ) p  /\  ps ) )  ->  ( z  = 
<. f ,  p >.  /\ 
th ) ) )
872eximdv 1882 . . . 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 3228 . . 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, 9eqsstrid 3201 . 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 4140 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 104    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   _Vcvv 2737   <.cop 3594   class class class wbr 4000   {copab 4060  (class class class)co 5868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4118
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-opab 4062
This theorem is referenced by: (None)
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