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Theorem dfoprab4pop 25037
Description: Class abstraction for operations in terms of class abstraction of ordered pairs. (A version of dfoprab4 6177 adapted to partial operations.) (Contributed by FL, 18-Apr-2010.)
Hypothesis
Ref Expression
dfoprab4pop.1  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
dfoprab4pop  |-  ( Rel 
R  ->  { <. w ,  z >.  |  ( w  e.  R  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( <. x ,  y
>.  e.  R  /\  ps ) } )
Distinct variable groups:    w, R, x, y, z    ph, x, y    ps, w
Allowed substitution hints:    ph( z, w)    ps( x, y, z)

Proof of Theorem dfoprab4pop
StepHypRef Expression
1 elrel 4789 . . . . . . 7  |-  ( ( Rel  R  /\  w  e.  R )  ->  E. x E. y  w  =  <. x ,  y >.
)
21adantrr 697 . . . . . 6  |-  ( ( Rel  R  /\  (
w  e.  R  /\  ph ) )  ->  E. x E. y  w  =  <. x ,  y >.
)
32ex 423 . . . . 5  |-  ( Rel 
R  ->  ( (
w  e.  R  /\  ph )  ->  E. x E. y  w  =  <. x ,  y >.
) )
43pm4.71rd 616 . . . 4  |-  ( Rel 
R  ->  ( (
w  e.  R  /\  ph )  <->  ( E. x E. y  w  =  <. x ,  y >.  /\  ( w  e.  R  /\  ph ) ) ) )
5 19.41vv 1843 . . . . 5  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  (
w  e.  R  /\  ph ) )  <->  ( E. x E. y  w  = 
<. x ,  y >.  /\  ( w  e.  R  /\  ph ) ) )
6 eleq1 2343 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( w  e.  R  <->  <. x ,  y
>.  e.  R ) )
7 dfoprab4pop.1 . . . . . . . 8  |-  ( w  =  <. x ,  y
>.  ->  ( ph  <->  ps )
)
86, 7anbi12d 691 . . . . . . 7  |-  ( w  =  <. x ,  y
>.  ->  ( ( w  e.  R  /\  ph ) 
<->  ( <. x ,  y
>.  e.  R  /\  ps ) ) )
98pm5.32i 618 . . . . . 6  |-  ( ( w  =  <. x ,  y >.  /\  (
w  e.  R  /\  ph ) )  <->  ( w  =  <. x ,  y
>.  /\  ( <. x ,  y >.  e.  R  /\  ps ) ) )
1092exbii 1570 . . . . 5  |-  ( E. x E. y ( w  =  <. x ,  y >.  /\  (
w  e.  R  /\  ph ) )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  ( <. x ,  y
>.  e.  R  /\  ps ) ) )
115, 10bitr3i 242 . . . 4  |-  ( ( E. x E. y  w  =  <. x ,  y >.  /\  (
w  e.  R  /\  ph ) )  <->  E. x E. y ( w  = 
<. x ,  y >.  /\  ( <. x ,  y
>.  e.  R  /\  ps ) ) )
124, 11syl6bb 252 . . 3  |-  ( Rel 
R  ->  ( (
w  e.  R  /\  ph )  <->  E. x E. y
( w  =  <. x ,  y >.  /\  ( <. x ,  y >.  e.  R  /\  ps )
) ) )
1312opabbidv 4082 . 2  |-  ( Rel 
R  ->  { <. w ,  z >.  |  ( w  e.  R  /\  ph ) }  =  { <. w ,  z >.  |  E. x E. y
( w  =  <. x ,  y >.  /\  ( <. x ,  y >.  e.  R  /\  ps )
) } )
14 dfoprab2 5895 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  (
<. x ,  y >.  e.  R  /\  ps ) }  =  { <. w ,  z >.  |  E. x E. y ( w  =  <. x ,  y
>.  /\  ( <. x ,  y >.  e.  R  /\  ps ) ) }
1513, 14syl6eqr 2333 1  |-  ( Rel 
R  ->  { <. w ,  z >.  |  ( w  e.  R  /\  ph ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( <. x ,  y
>.  e.  R  /\  ps ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643   {copab 4076   Rel wrel 4694   {coprab 5859
This theorem is referenced by:  fnovpop  25038
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-oprab 5862
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