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Theorem elxp5 5243
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5242 when the double intersection does not create class existence problems (caused by int0 3957). (Contributed by NM, 1-Aug-2004.)
Assertion
Ref Expression
elxp5  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )

Proof of Theorem elxp5
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp 4788 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
2 sneq 3727 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
32rneqd 4988 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
43unieqd 3919 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
5 vex 2867 . . . . . . . . . . 11  |-  x  e. 
_V
6 vex 2867 . . . . . . . . . . 11  |-  y  e. 
_V
75, 6op2nda 5239 . . . . . . . . . 10  |-  U. ran  {
<. x ,  y >. }  =  y
84, 7syl6req 2407 . . . . . . . . 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 1582 . . . . 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 4297 . . . . . . . 8  |-  { A }  e.  _V
1514rnex 5024 . . . . . . 7  |-  ran  { A }  e.  _V
1615uniex 4598 . . . . . 6  |-  U. ran  { A }  e.  _V
17 opeq2 3878 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
1817eqeq2d 2369 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
19 eleq1 2418 . . . . . . . 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 2899 . . . . 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 inteq 3946 . . . . . . . 8  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| A  =  |^| <. x ,  U. ran  { A } >. )
2524inteqd 3948 . . . . . . 7  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| |^| A  =  |^| |^|
<. x ,  U. ran  { A } >. )
265, 16op1stb 4651 . . . . . . 7  |-  |^| |^| <. x ,  U. ran  { A } >.  =  x
2725, 26syl6req 2407 . . . . . 6  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  x  =  |^| |^| A
)
2827pm4.71ri 614 . . . . 5  |-  ( A  =  <. x ,  U. ran  { A } >.  <->  (
x  =  |^| |^| A  /\  A  =  <. x ,  U. ran  { A } >. ) )
2928anbi1i 676 . . . 4  |-  ( ( A  =  <. x ,  U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) )  <->  ( (
x  =  |^| |^| A  /\  A  =  <. x ,  U. ran  { A } >. )  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
30 anass 630 . . . 4  |-  ( ( ( x  =  |^| |^| A  /\  A  = 
<. x ,  U. ran  { A } >. )  /\  ( x  e.  B  /\  U. ran  { A }  e.  C )
)  <->  ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
3123, 29, 303bitri 262 . . 3  |-  ( E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) )  <->  ( x  =  |^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
3231exbii 1582 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  E. x ( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) ) )
33 eleq1 2418 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( x  e.  _V  <->  |^|
|^| A  e.  _V ) )
345, 33mpbii 202 . . . . 5  |-  ( x  =  |^| |^| A  ->  |^| |^| A  e.  _V )
3534adantr 451 . . . 4  |-  ( ( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )  ->  |^| |^| A  e.  _V )
3635exlimiv 1634 . . 3  |-  ( E. x ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )  ->  |^| |^| A  e.  _V )
37 elex 2872 . . . 4  |-  ( |^| |^| A  e.  B  ->  |^| |^| A  e.  _V )
3837ad2antrl 708 . . 3  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  |^| |^| A  e.  _V )
39 opeq1 3877 . . . . . 6  |-  ( x  =  |^| |^| A  -> 
<. x ,  U. ran  { A } >.  =  <. |^|
|^| A ,  U. ran  { A } >. )
4039eqeq2d 2369 . . . . 5  |-  ( x  =  |^| |^| A  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. |^|
|^| A ,  U. ran  { A } >. ) )
41 eleq1 2418 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( x  e.  B  <->  |^|
|^| A  e.  B
) )
4241anbi1d 685 . . . . 5  |-  ( x  =  |^| |^| A  ->  ( ( x  e.  B  /\  U. ran  { A }  e.  C
)  <->  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
4340, 42anbi12d 691 . . . 4  |-  ( x  =  |^| |^| A  ->  ( ( A  = 
<. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
4443ceqsexgv 2976 . . 3  |-  ( |^| |^| A  e.  _V  ->  ( E. x ( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )  <-> 
( A  =  <. |^|
|^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
4536, 38, 44pm5.21nii 342 . 2  |-  ( E. x ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
461, 32, 453bitri 262 1  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   _Vcvv 2864   {csn 3716   <.cop 3719   U.cuni 3908   |^|cint 3943    X. cxp 4769   ran crn 4772
This theorem is referenced by:  xpnnenOLD  12585
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-int 3944  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-dm 4781  df-rn 4782
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