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Theorem inopab 5005
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 5001 . . 3  |-  Rel  { <. x ,  y >.  |  ph }
2 relin1 4992 . . 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 5001 . 2  |-  Rel  { <. x ,  y >.  |  ( ph  /\  ps ) }
5 sban 2139 . . . 4  |-  ( [ w  /  y ] ( [ z  /  x ] ph  /\  [
z  /  x ] ps )  <->  ( [ w  /  y ] [
z  /  x ] ph  /\  [ w  / 
y ] [ z  /  x ] ps ) )
6 sban 2139 . . . . 5  |-  ( [ z  /  x ]
( ph  /\  ps )  <->  ( [ z  /  x ] ph  /\  [ z  /  x ] ps ) )
76sbbii 1665 . . . 4  |-  ( [ w  /  y ] [ z  /  x ] ( ph  /\  ps )  <->  [ w  /  y ] ( [ z  /  x ] ph  /\ 
[ z  /  x ] ps ) )
8 opelopabsbOLD 4463 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
9 opelopabsbOLD 4463 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps }  <->  [ w  /  y ] [
z  /  x ] ps )
108, 9anbi12i 679 . . . 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 270 . . 3  |-  ( (
<. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  /\  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps } )  <->  [ w  /  y ] [ z  /  x ] ( ph  /\  ps ) )
12 elin 3530 . . 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 4463 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ( ph  /\ 
ps ) }  <->  [ w  /  y ] [
z  /  x ]
( ph  /\  ps )
)
1411, 12, 133bitr4i 269 . 2  |-  ( <.
z ,  w >.  e.  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  (
ph  /\  ps ) } )
153, 4, 14eqrelriiv 4970 1  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652   [wsb 1658    e. wcel 1725    i^i cin 3319   <.cop 3817   {copab 4265   Rel wrel 4883
This theorem is referenced by:  inxp  5007  resopab  5187  fndmin  5837  wemapwe  7654  frgpuplem  15404  ltbwe  16533  opsrtoslem1  16544  pjfval2  16936  lgsquadlem3  21140  dnwech  27123  fgraphopab  27506
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884  df-rel 4885
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