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Theorem elxp4 5357
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5358, elxp6 6378, and elxp7 6379. (Contributed by NM, 17-Feb-2004.)
Assertion
Ref Expression
elxp4  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )

Proof of Theorem elxp4
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4895 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
2 sneq 3825 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
32rneqd 5097 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
43unieqd 4026 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
5 vex 2959 . . . . . . . . . . 11  |-  x  e. 
_V
6 vex 2959 . . . . . . . . . . 11  |-  y  e. 
_V
75, 6op2nda 5354 . . . . . . . . . 10  |-  U. ran  {
<. x ,  y >. }  =  y
84, 7syl6req 2485 . . . . . . . . 9  |-  ( A  =  <. x ,  y
>.  ->  y  =  U. ran  { A } )
98pm4.71ri 615 . . . . . . . 8  |-  ( A  =  <. x ,  y
>. 
<->  ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. ) )
109anbi1i 677 . . . . . . 7  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( ( y  =  U. ran  { A }  /\  A  = 
<. x ,  y >.
)  /\  ( x  e.  B  /\  y  e.  C ) ) )
11 anass 631 . . . . . . 7  |-  ( ( ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. )  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
1210, 11bitri 241 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
1312exbii 1592 . . . . 5  |-  ( E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) )  <->  E. y
( y  =  U. ran  { A }  /\  ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
) ) )
14 snex 4405 . . . . . . . 8  |-  { A }  e.  _V
1514rnex 5133 . . . . . . 7  |-  ran  { A }  e.  _V
1615uniex 4705 . . . . . 6  |-  U. ran  { A }  e.  _V
17 opeq2 3985 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
1817eqeq2d 2447 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
19 eleq1 2496 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  ( y  e.  C  <->  U. ran  { A }  e.  C
) )
2019anbi2d 685 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( ( x  e.  B  /\  y  e.  C )  <->  ( x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
2118, 20anbi12d 692 . . . . . 6  |-  ( y  =  U. ran  { A }  ->  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( A  = 
<. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) ) )
2216, 21ceqsexv 2991 . . . . 5  |-  ( E. y ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )  <->  ( A  =  <. x ,  U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C )
) )
2313, 22bitri 241 . . . 4  |-  ( E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) )  <->  ( A  =  <. x ,  U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C )
) )
24 sneq 3825 . . . . . . . . 9  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  { A }  =  { <. x ,  U. ran  { A } >. } )
2524dmeqd 5072 . . . . . . . 8  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  dom  { A }  =  dom  { <. x ,  U. ran  { A } >. } )
2625unieqd 4026 . . . . . . 7  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  U. dom  { A }  =  U. dom  { <. x ,  U. ran  { A } >. } )
275, 16op1sta 5351 . . . . . . 7  |-  U. dom  {
<. x ,  U. ran  { A } >. }  =  x
2826, 27syl6req 2485 . . . . . 6  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  x  =  U. dom  { A } )
2928pm4.71ri 615 . . . . 5  |-  ( A  =  <. x ,  U. ran  { A } >.  <->  (
x  =  U. dom  { A }  /\  A  =  <. x ,  U. ran  { A } >. ) )
3029anbi1i 677 . . . 4  |-  ( ( A  =  <. x ,  U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) )  <->  ( (
x  =  U. dom  { A }  /\  A  =  <. x ,  U. ran  { A } >. )  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )
31 anass 631 . . . 4  |-  ( ( ( x  =  U. dom  { A }  /\  A  =  <. x , 
U. ran  { A } >. )  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) )  <->  ( x  =  U. dom  { A }  /\  ( A  = 
<. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) ) )
3223, 30, 313bitri 263 . . 3  |-  ( E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) )  <->  ( x  =  U. dom  { A }  /\  ( A  = 
<. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) ) )
3332exbii 1592 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  E. x ( x  =  U. dom  { A }  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
3414dmex 5132 . . . 4  |-  dom  { A }  e.  _V
3534uniex 4705 . . 3  |-  U. dom  { A }  e.  _V
36 opeq1 3984 . . . . 5  |-  ( x  =  U. dom  { A }  ->  <. x ,  U. ran  { A } >.  =  <. U. dom  { A } ,  U. ran  { A } >. )
3736eqeq2d 2447 . . . 4  |-  ( x  =  U. dom  { A }  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. U. dom  { A } ,  U. ran  { A } >. ) )
38 eleq1 2496 . . . . 5  |-  ( x  =  U. dom  { A }  ->  ( x  e.  B  <->  U. dom  { A }  e.  B
) )
3938anbi1d 686 . . . 4  |-  ( x  =  U. dom  { A }  ->  ( ( x  e.  B  /\  U.
ran  { A }  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
4037, 39anbi12d 692 . . 3  |-  ( x  =  U. dom  { A }  ->  ( ( A  =  <. x ,  U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
4135, 40ceqsexv 2991 . 2  |-  ( E. x ( x  = 
U. dom  { A }  /\  ( A  = 
<. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )  <-> 
( A  =  <. U.
dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C )
) )
421, 33, 413bitri 263 1  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {csn 3814   <.cop 3817   U.cuni 4015    X. cxp 4876   dom cdm 4878   ran crn 4879
This theorem is referenced by:  elxp6  6378  xpdom2  7203
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-13 1727  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  ax-un 4701
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-eu 2285  df-mo 2286  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-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889
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