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Theorem inopab 4816
Description: Intersection of two ordered pair class abstractions. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
inopab  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  /\ 
ps ) }
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem inopab
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 4812 . . 3  |-  Rel  { <. x ,  y >.  |  ph }
2 relin1 4803 . . 3  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  ->  Rel  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } ) )
31, 2ax-mp 8 . 2  |-  Rel  ( { <. x ,  y
>.  |  ph }  i^i  {
<. x ,  y >.  |  ps } )
4 relopab 4812 . 2  |-  Rel  { <. x ,  y >.  |  ( ph  /\  ps ) }
5 sban 2009 . . . 4  |-  ( [ w  /  y ] ( [ z  /  x ] ph  /\  [
z  /  x ] ps )  <->  ( [ w  /  y ] [
z  /  x ] ph  /\  [ w  / 
y ] [ z  /  x ] ps ) )
6 sban 2009 . . . . 5  |-  ( [ z  /  x ]
( ph  /\  ps )  <->  ( [ z  /  x ] ph  /\  [ z  /  x ] ps ) )
76sbbii 1634 . . . 4  |-  ( [ w  /  y ] [ z  /  x ] ( ph  /\  ps )  <->  [ w  /  y ] ( [ z  /  x ] ph  /\ 
[ z  /  x ] ps ) )
8 opelopabsbOLD 4273 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
9 opelopabsbOLD 4273 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps }  <->  [ w  /  y ] [
z  /  x ] ps )
108, 9anbi12i 678 . . . 4  |-  ( (
<. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  /\  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps } )  <-> 
( [ w  / 
y ] [ z  /  x ] ph  /\ 
[ w  /  y ] [ z  /  x ] ps ) )
115, 7, 103bitr4ri 269 . . 3  |-  ( (
<. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  /\  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps } )  <->  [ w  /  y ] [ z  /  x ] ( ph  /\  ps ) )
12 elin 3358 . . 3  |-  ( <.
z ,  w >.  e.  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  <->  ( <. z ,  w >.  e.  { <. x ,  y >.  |  ph }  /\  <. z ,  w >.  e.  { <. x ,  y >.  |  ps } ) )
13 opelopabsbOLD 4273 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ( ph  /\ 
ps ) }  <->  [ w  /  y ] [
z  /  x ]
( ph  /\  ps )
)
1411, 12, 133bitr4i 268 . 2  |-  ( <.
z ,  w >.  e.  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  (
ph  /\  ps ) } )
153, 4, 14eqrelriiv 4781 1  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623   [wsb 1629    e. wcel 1684    i^i cin 3151   <.cop 3643   {copab 4076   Rel wrel 4694
This theorem is referenced by:  inxp  4818  resopab  4996  fndmin  5632  wemapwe  7400  frgpuplem  15081  ltbwe  16214  opsrtoslem1  16225  pjfval2  16609  lgsquadlem3  20595  inposet  25278  domncnt  25282  ranncnt  25283  dnwech  27145  fgraphopab  27529
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-ral 2548  df-rex 2549  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
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