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Theorem inopab 4853
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 4849 . . 3  |-  Rel  { <. x ,  y >.  |  ph }
2 relin1 4840 . . 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 4849 . 2  |-  Rel  { <. x ,  y >.  |  ( ph  /\  ps ) }
5 sban 2041 . . . 4  |-  ( [ w  /  y ] ( [ z  /  x ] ph  /\  [
z  /  x ] ps )  <->  ( [ w  /  y ] [
z  /  x ] ph  /\  [ w  / 
y ] [ z  /  x ] ps ) )
6 sban 2041 . . . . 5  |-  ( [ z  /  x ]
( ph  /\  ps )  <->  ( [ z  /  x ] ph  /\  [ z  /  x ] ps ) )
76sbbii 1644 . . . 4  |-  ( [ w  /  y ] [ z  /  x ] ( ph  /\  ps )  <->  [ w  /  y ] ( [ z  /  x ] ph  /\ 
[ z  /  x ] ps ) )
8 opelopabsbOLD 4310 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
9 opelopabsbOLD 4310 . . . . 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 3392 . . 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 4310 . . 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 4818 1  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1633   [wsb 1639    e. wcel 1701    i^i cin 3185   <.cop 3677   {copab 4113   Rel wrel 4731
This theorem is referenced by:  inxp  4855  resopab  5033  fndmin  5670  wemapwe  7445  frgpuplem  15130  ltbwe  16263  opsrtoslem1  16274  pjfval2  16665  lgsquadlem3  20648  dnwech  26293  fgraphopab  26677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-opab 4115  df-xp 4732  df-rel 4733
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