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Theorem elxp5 6696
Description: Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 6695 when the double intersection does not create class existence problems (caused by int0 4212). (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 4813 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
2 sneq 3951 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
32rneqd 5024 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
43unieqd 4172 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
5 vex 3025 . . . . . . . . . . 11  |-  x  e. 
_V
6 vex 3025 . . . . . . . . . . 11  |-  y  e. 
_V
75, 6op2nda 5283 . . . . . . . . . 10  |-  U. ran  {
<. x ,  y >. }  =  y
84, 7syl6req 2479 . . . . . . . . 9  |-  ( A  =  <. x ,  y
>.  ->  y  =  U. ran  { A } )
98pm4.71ri 637 . . . . . . . 8  |-  ( A  =  <. x ,  y
>. 
<->  ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. ) )
109anbi1i 699 . . . . . . 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 653 . . . . . . 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 252 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
1312exbii 1712 . . . . 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 4605 . . . . . . . 8  |-  { A }  e.  _V
1514rnex 6685 . . . . . . 7  |-  ran  { A }  e.  _V
1615uniex 6545 . . . . . 6  |-  U. ran  { A }  e.  _V
17 opeq2 4131 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
1817eqeq2d 2438 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
19 eleq1 2494 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  ( y  e.  C  <->  U. ran  { A }  e.  C
) )
2019anbi2d 708 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( ( x  e.  B  /\  y  e.  C )  <->  ( x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
2118, 20anbi12d 715 . . . . . 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 3060 . . . . 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 252 . . . 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 4201 . . . . . . . 8  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| A  =  |^| <. x ,  U. ran  { A } >. )
2524inteqd 4203 . . . . . . 7  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| |^| A  =  |^| |^|
<. x ,  U. ran  { A } >. )
265, 16op1stb 4634 . . . . . . 7  |-  |^| |^| <. x ,  U. ran  { A } >.  =  x
2725, 26syl6req 2479 . . . . . 6  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  x  =  |^| |^| A
)
2827pm4.71ri 637 . . . . 5  |-  ( A  =  <. x ,  U. ran  { A } >.  <->  (
x  =  |^| |^| A  /\  A  =  <. x ,  U. ran  { A } >. ) )
2928anbi1i 699 . . . 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 653 . . . 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 274 . . 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 1712 . 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 eqvisset 3030 . . . . 5  |-  ( x  =  |^| |^| A  ->  |^| |^| A  e.  _V )
3433adantr 466 . . . 4  |-  ( ( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )  ->  |^| |^| A  e.  _V )
3534exlimiv 1770 . . 3  |-  ( E. x ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )  ->  |^| |^| A  e.  _V )
36 elex 3031 . . . 4  |-  ( |^| |^| A  e.  B  ->  |^| |^| A  e.  _V )
3736ad2antrl 732 . . 3  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  |^| |^| A  e.  _V )
38 opeq1 4130 . . . . . 6  |-  ( x  =  |^| |^| A  -> 
<. x ,  U. ran  { A } >.  =  <. |^|
|^| A ,  U. ran  { A } >. )
3938eqeq2d 2438 . . . . 5  |-  ( x  =  |^| |^| A  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. |^|
|^| A ,  U. ran  { A } >. ) )
40 eleq1 2494 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( x  e.  B  <->  |^|
|^| A  e.  B
) )
4140anbi1d 709 . . . . 5  |-  ( x  =  |^| |^| A  ->  ( ( x  e.  B  /\  U. ran  { A }  e.  C
)  <->  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
4239, 41anbi12d 715 . . . 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
) ) ) )
4342ceqsexgv 3146 . . 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
) ) ) )
4435, 37, 43pm5.21nii 354 . 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
) ) )
451, 32, 443bitri 274 1  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   _Vcvv 3022   {csn 3941   <.cop 3947   U.cuni 4162   |^|cint 4198    X. cxp 4794   ran crn 4797
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-8 1874  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  ax-un 6541
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-eu 2280  df-mo 2281  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-uni 4163  df-int 4199  df-br 4367  df-opab 4426  df-xp 4802  df-rel 4803  df-cnv 4804  df-dm 4806  df-rn 4807
This theorem is referenced by: (None)
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