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Theorem elxp4 5176
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 5177, elxp6 6167, and elxp7 6168. (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 4722 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
2 sneq 3664 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
32rneqd 4922 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
43unieqd 3854 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
5 vex 2804 . . . . . . . . . . 11  |-  x  e. 
_V
6 vex 2804 . . . . . . . . . . 11  |-  y  e. 
_V
75, 6op2nda 5173 . . . . . . . . . 10  |-  U. ran  {
<. x ,  y >. }  =  y
84, 7syl6req 2345 . . . . . . . . 9  |-  ( A  =  <. x ,  y
>.  ->  y  =  U. ran  { A } )
98pm4.71ri 614 . . . . . . . 8  |-  ( A  =  <. x ,  y
>. 
<->  ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. ) )
109anbi1i 676 . . . . . . 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 630 . . . . . . 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 240 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
1312exbii 1572 . . . . 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 4232 . . . . . . . 8  |-  { A }  e.  _V
1514rnex 4958 . . . . . . 7  |-  ran  { A }  e.  _V
1615uniex 4532 . . . . . 6  |-  U. ran  { A }  e.  _V
17 opeq2 3813 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
1817eqeq2d 2307 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
19 eleq1 2356 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  ( y  e.  C  <->  U. ran  { A }  e.  C
) )
2019anbi2d 684 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( ( x  e.  B  /\  y  e.  C )  <->  ( x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
2118, 20anbi12d 691 . . . . . 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 2836 . . . . 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 240 . . . 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 3664 . . . . . . . . 9  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  { A }  =  { <. x ,  U. ran  { A } >. } )
2524dmeqd 4897 . . . . . . . 8  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  dom  { A }  =  dom  { <. x ,  U. ran  { A } >. } )
2625unieqd 3854 . . . . . . 7  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  U. dom  { A }  =  U. dom  { <. x ,  U. ran  { A } >. } )
275, 16op1sta 5170 . . . . . . 7  |-  U. dom  {
<. x ,  U. ran  { A } >. }  =  x
2826, 27syl6req 2345 . . . . . 6  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  x  =  U. dom  { A } )
2928pm4.71ri 614 . . . . 5  |-  ( A  =  <. x ,  U. ran  { A } >.  <->  (
x  =  U. dom  { A }  /\  A  =  <. x ,  U. ran  { A } >. ) )
3029anbi1i 676 . . . 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 630 . . . 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 262 . . 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 1572 . 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 4957 . . . 4  |-  dom  { A }  e.  _V
3534uniex 4532 . . 3  |-  U. dom  { A }  e.  _V
36 opeq1 3812 . . . . 5  |-  ( x  =  U. dom  { A }  ->  <. x ,  U. ran  { A } >.  =  <. U. dom  { A } ,  U. ran  { A } >. )
3736eqeq2d 2307 . . . 4  |-  ( x  =  U. dom  { A }  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. U. dom  { A } ,  U. ran  { A } >. ) )
38 eleq1 2356 . . . . 5  |-  ( x  =  U. dom  { A }  ->  ( x  e.  B  <->  U. dom  { A }  e.  B
) )
3938anbi1d 685 . . . 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 691 . . 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 2836 . 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 262 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 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656   U.cuni 3843    X. cxp 4703   dom cdm 4705   ran crn 4706
This theorem is referenced by:  elxp6  6167  xpdom2  6973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
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