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Theorem opabbrexOLD 6348
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.) (Revised by BJ/AV, 20-Jun-2019.) Obsolete version of opabbrex 6347 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
opabbrexOLD  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { <. x ,  y >.  |  ( x R y  /\  ps ) }  e.  _V )

Proof of Theorem opabbrexOLD
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-opab 4483 . . 3  |-  { <. x ,  y >.  |  ph }  =  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }
2 simpr 462 . . 3  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { <. x ,  y >.  |  ph }  e.  V )
31, 2syl5eqelr 2512 . 2  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) }  e.  V
)
4 df-opab 4483 . . 3  |-  { <. x ,  y >.  |  ( x R y  /\  ps ) }  =  {
z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ( x R y  /\  ps ) ) }
5 pm3.41 561 . . . . . . . 8  |-  ( ( x R y  ->  ph )  ->  ( ( x R y  /\  ps )  ->  ph )
)
65anim2d 567 . . . . . . 7  |-  ( ( x R y  ->  ph )  ->  ( ( z  =  <. x ,  y >.  /\  (
x R y  /\  ps ) )  ->  (
z  =  <. x ,  y >.  /\  ph ) ) )
76aleximi 1698 . . . . . 6  |-  ( A. y ( x R y  ->  ph )  -> 
( E. y ( z  =  <. x ,  y >.  /\  (
x R y  /\  ps ) )  ->  E. y
( z  =  <. x ,  y >.  /\  ph ) ) )
87aleximi 1698 . . . . 5  |-  ( A. x A. y ( x R y  ->  ph )  ->  ( E. x E. y ( z  = 
<. x ,  y >.  /\  ( x R y  /\  ps ) )  ->  E. x E. y
( z  =  <. x ,  y >.  /\  ph ) ) )
98adantr 466 . . . 4  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  ( E. x E. y ( z  =  <. x ,  y
>.  /\  ( x R y  /\  ps )
)  ->  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) ) )
109ss2abdv 3534 . . 3  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { z  |  E. x E. y
( z  =  <. x ,  y >.  /\  (
x R y  /\  ps ) ) }  C_  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) } )
114, 10syl5eqss 3508 . 2  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { <. x ,  y >.  |  ( x R y  /\  ps ) }  C_  { z  |  E. x E. y ( z  = 
<. x ,  y >.  /\  ph ) } )
123, 11ssexd 4571 1  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { <. x ,  y >.  |  ( x R y  /\  ps ) }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435    = wceq 1437   E.wex 1657    e. wcel 1872   {cab 2407   _Vcvv 3080   <.cop 4004   class class class wbr 4423   {copab 4481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-in 3443  df-ss 3450  df-opab 4483
This theorem is referenced by: (None)
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