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Theorem inopab 4526
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 4522 . . 3  |-  Rel  { <. x ,  y >.  |  ph }
2 relin1 4513 . . 3  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  ->  Rel  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } ) )
31, 2ax-mp 7 . 2  |-  Rel  ( { <. x ,  y
>.  |  ph }  i^i  {
<. x ,  y >.  |  ps } )
4 relopab 4522 . 2  |-  Rel  { <. x ,  y >.  |  ( ph  /\  ps ) }
5 sban 1872 . . . 4  |-  ( [ w  /  y ] ( [ z  /  x ] ph  /\  [
z  /  x ] ps )  <->  ( [ w  /  y ] [
z  /  x ] ph  /\  [ w  / 
y ] [ z  /  x ] ps ) )
6 sban 1872 . . . . 5  |-  ( [ z  /  x ]
( ph  /\  ps )  <->  ( [ z  /  x ] ph  /\  [ z  /  x ] ps ) )
76sbbii 1690 . . . 4  |-  ( [ w  /  y ] [ z  /  x ] ( ph  /\  ps )  <->  [ w  /  y ] ( [ z  /  x ] ph  /\ 
[ z  /  x ] ps ) )
8 opelopabsbALT 4050 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
9 opelopabsbALT 4050 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps }  <->  [ w  /  y ] [
z  /  x ] ps )
108, 9anbi12i 448 . . . 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 211 . . 3  |-  ( (
<. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  /\  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps } )  <->  [ w  /  y ] [ z  /  x ] ( ph  /\  ps ) )
12 elin 3167 . . 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 opelopabsbALT 4050 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ( ph  /\ 
ps ) }  <->  [ w  /  y ] [
z  /  x ]
( ph  /\  ps )
)
1411, 12, 133bitr4i 210 . 2  |-  ( <.
z ,  w >.  e.  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  (
ph  /\  ps ) } )
153, 4, 14eqrelriiv 4490 1  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   [wsb 1687    i^i cin 2983   <.cop 3425   {copab 3864   Rel wrel 4406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-opab 3866  df-xp 4407  df-rel 4408
This theorem is referenced by:  inxp  4528  resopab  4713  cnvin  4793  fndmin  5351  enq0enq  6893
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