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Theorem opabbrex 6347
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.) (Proof shortened by OpenAI, 25-Mar-2020.)
Assertion
Ref Expression
opabbrex  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { <. x ,  y >.  |  ( x R y  /\  ps ) }  e.  _V )

Proof of Theorem opabbrex
StepHypRef Expression
1 simpr 462 . 2  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { <. x ,  y >.  |  ph }  e.  V )
2 pm3.41 561 . . . . 5  |-  ( ( x R y  ->  ph )  ->  ( ( x R y  /\  ps )  ->  ph )
)
322alimi 1679 . . . 4  |-  ( A. x A. y ( x R y  ->  ph )  ->  A. x A. y
( ( x R y  /\  ps )  ->  ph ) )
43adantr 466 . . 3  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  A. x A. y ( ( x R y  /\  ps )  ->  ph ) )
5 ssopab2 4746 . . 3  |-  ( A. x A. y ( ( x R y  /\  ps )  ->  ph )  ->  { <. x ,  y
>.  |  ( x R y  /\  ps ) }  C_  { <. x ,  y >.  |  ph } )
64, 5syl 17 . 2  |-  ( ( A. x A. y
( x R y  ->  ph )  /\  { <. x ,  y >.  |  ph }  e.  V
)  ->  { <. x ,  y >.  |  ( x R y  /\  ps ) }  C_  { <. x ,  y >.  |  ph } )
71, 6ssexd 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    e. wcel 1872   _Vcvv 3080    C_ wss 3436   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:  sprmpt2d  6981  wlkres  25248
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