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Theorem inxp 4929
Description: The intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
inxp  |-  ( ( A  X.  B )  i^i  ( C  X.  D ) )  =  ( ( A  i^i  C )  X.  ( B  i^i  D ) )

Proof of Theorem inxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopab 4927 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  i^i  {
<. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) } )  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ( x  e.  C  /\  y  e.  D ) ) }
2 an4 831 . . . . 5  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( x  e.  C  /\  y  e.  D ) )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  ( y  e.  B  /\  y  e.  D
) ) )
3 elin 3592 . . . . . 6  |-  ( x  e.  ( A  i^i  C )  <->  ( x  e.  A  /\  x  e.  C ) )
4 elin 3592 . . . . . 6  |-  ( y  e.  ( B  i^i  D )  <->  ( y  e.  B  /\  y  e.  D ) )
53, 4anbi12i 701 . . . . 5  |-  ( ( x  e.  ( A  i^i  C )  /\  y  e.  ( B  i^i  D ) )  <->  ( (
x  e.  A  /\  x  e.  C )  /\  ( y  e.  B  /\  y  e.  D
) ) )
62, 5bitr4i 255 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  ( x  e.  C  /\  y  e.  D ) )  <->  ( x  e.  ( A  i^i  C
)  /\  y  e.  ( B  i^i  D ) ) )
76opabbii 4431 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  ( x  e.  C  /\  y  e.  D ) ) }  =  { <. x ,  y >.  |  ( x  e.  ( A  i^i  C )  /\  y  e.  ( B  i^i  D ) ) }
81, 7eqtri 2450 . 2  |-  ( {
<. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  i^i  {
<. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) } )  =  { <. x ,  y >.  |  ( x  e.  ( A  i^i  C )  /\  y  e.  ( B  i^i  D ) ) }
9 df-xp 4802 . . 3  |-  ( A  X.  B )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  B ) }
10 df-xp 4802 . . 3  |-  ( C  X.  D )  =  { <. x ,  y
>.  |  ( x  e.  C  /\  y  e.  D ) }
119, 10ineq12i 3605 . 2  |-  ( ( A  X.  B )  i^i  ( C  X.  D ) )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  B ) }  i^i  { <. x ,  y >.  |  ( x  e.  C  /\  y  e.  D ) } )
12 df-xp 4802 . 2  |-  ( ( A  i^i  C )  X.  ( B  i^i  D ) )  =  { <. x ,  y >.  |  ( x  e.  ( A  i^i  C
)  /\  y  e.  ( B  i^i  D ) ) }
138, 11, 123eqtr4i 2460 1  |-  ( ( A  X.  B )  i^i  ( C  X.  D ) )  =  ( ( A  i^i  C )  X.  ( B  i^i  D ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1872    i^i cin 3378   {copab 4424    X. cxp 4794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-opab 4426  df-xp 4802  df-rel 4803
This theorem is referenced by:  xpindi  4930  xpindir  4931  dmxpin  5017  xpssres  5101  xpdisj1  5220  xpdisj2  5221  imainrect  5240  xpima  5241  curry1  6843  curry2  6846  fpar  6855  marypha1lem  7900  fpwwe2lem13  9018  hashxplem  12553  sscres  15671  gsumxp  17551  pjfval  19211  pjpm  19213  txbas  20524  txcls  20561  txrest  20588  trust  21186  ressuss  21220  trcfilu  21251  metreslem  21319  ressxms  21482  ressms  21483  mbfmcst  29033  0rrv  29236  poimirlem26  31873  xpheOLD  36290
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