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Theorem inopab 4671
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 4666 . . 3  |-  Rel  { <. x ,  y >.  |  ph }
2 relin1 4657 . . 3  |-  ( Rel 
{ <. x ,  y
>.  |  ph }  ->  Rel  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } ) )
31, 2ax-mp 5 . 2  |-  Rel  ( { <. x ,  y
>.  |  ph }  i^i  {
<. x ,  y >.  |  ps } )
4 relopab 4666 . 2  |-  Rel  { <. x ,  y >.  |  ( ph  /\  ps ) }
5 sban 1928 . . . 4  |-  ( [ w  /  y ] ( [ z  /  x ] ph  /\  [
z  /  x ] ps )  <->  ( [ w  /  y ] [
z  /  x ] ph  /\  [ w  / 
y ] [ z  /  x ] ps ) )
6 sban 1928 . . . . 5  |-  ( [ z  /  x ]
( ph  /\  ps )  <->  ( [ z  /  x ] ph  /\  [ z  /  x ] ps ) )
76sbbii 1738 . . . 4  |-  ( [ w  /  y ] [ z  /  x ] ( ph  /\  ps )  <->  [ w  /  y ] ( [ z  /  x ] ph  /\ 
[ z  /  x ] ps ) )
8 opelopabsbALT 4181 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  <->  [ w  /  y ] [
z  /  x ] ph )
9 opelopabsbALT 4181 . . . . 5  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps }  <->  [ w  /  y ] [
z  /  x ] ps )
108, 9anbi12i 455 . . . 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 212 . . 3  |-  ( (
<. z ,  w >.  e. 
{ <. x ,  y
>.  |  ph }  /\  <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ps } )  <->  [ w  /  y ] [ z  /  x ] ( ph  /\  ps ) )
12 elin 3259 . . 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 4181 . . 3  |-  ( <.
z ,  w >.  e. 
{ <. x ,  y
>.  |  ( ph  /\ 
ps ) }  <->  [ w  /  y ] [
z  /  x ]
( ph  /\  ps )
)
1411, 12, 133bitr4i 211 . 2  |-  ( <.
z ,  w >.  e.  ( { <. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  <->  <. z ,  w >.  e.  { <. x ,  y >.  |  (
ph  /\  ps ) } )
153, 4, 14eqrelriiv 4633 1  |-  ( {
<. x ,  y >.  |  ph }  i^i  { <. x ,  y >.  |  ps } )  =  { <. x ,  y
>.  |  ( ph  /\ 
ps ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1331    e. wcel 1480   [wsb 1735    i^i cin 3070   <.cop 3530   {copab 3988   Rel wrel 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-xp 4545  df-rel 4546
This theorem is referenced by:  inxp  4673  resopab  4863  cnvin  4946  fndmin  5527  enq0enq  7246
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