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Theorem elxp5 5097
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5096 when the double intersection does not create class existence problems (caused by int0 3843). (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 elex 2741 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 elex 2741 . . . 4  |-  ( |^| |^| A  e.  B  ->  |^| |^| A  e.  _V )
3 elex 2741 . . . 4  |-  ( U. ran  { A }  e.  C  ->  U. ran  { A }  e.  _V )
42, 3anim12i 336 . . 3  |-  ( (
|^| |^| A  e.  B  /\  U. ran  { A }  e.  C )  ->  ( |^| |^| A  e.  _V  /\  U. ran  { A }  e.  _V ) )
5 opexg 4211 . . . . 5  |-  ( (
|^| |^| A  e.  _V  /\ 
U. ran  { A }  e.  _V )  -> 
<. |^| |^| A ,  U. ran  { A } >.  e. 
_V )
65adantl 275 . . . 4  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  _V  /\  U.
ran  { A }  e.  _V ) )  ->  <. |^| |^| A ,  U. ran  { A } >.  e.  _V )
7 eleq1 2233 . . . . 5  |-  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  -> 
( A  e.  _V  <->  <. |^| |^| A ,  U. ran  { A } >.  e. 
_V ) )
87adantr 274 . . . 4  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  _V  /\  U.
ran  { A }  e.  _V ) )  ->  ( A  e.  _V  <->  <. |^| |^| A ,  U. ran  { A } >.  e.  _V )
)
96, 8mpbird 166 . . 3  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  _V  /\  U.
ran  { A }  e.  _V ) )  ->  A  e.  _V )
104, 9sylan2 284 . 2  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  A  e.  _V )
11 elxp 4626 . . . 4  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
12 sneq 3592 . . . . . . . . . . . . . 14  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
1312rneqd 4838 . . . . . . . . . . . . 13  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
1413unieqd 3805 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
15 vex 2733 . . . . . . . . . . . . 13  |-  x  e. 
_V
16 vex 2733 . . . . . . . . . . . . 13  |-  y  e. 
_V
1715, 16op2nda 5093 . . . . . . . . . . . 12  |-  U. ran  {
<. x ,  y >. }  =  y
1814, 17eqtr2di 2220 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  y  =  U. ran  { A } )
1918pm4.71ri 390 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>. 
<->  ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. ) )
2019anbi1i 455 . . . . . . . . 9  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( ( y  =  U. ran  { A }  /\  A  = 
<. x ,  y >.
)  /\  ( x  e.  B  /\  y  e.  C ) ) )
21 anass 399 . . . . . . . . 9  |-  ( ( ( 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
) ) ) )
2220, 21bitri 183 . . . . . . . 8  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
2322exbii 1598 . . . . . . 7  |-  ( 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 )
) ) )
24 snexg 4168 . . . . . . . . . 10  |-  ( A  e.  _V  ->  { A }  e.  _V )
25 rnexg 4874 . . . . . . . . . 10  |-  ( { A }  e.  _V  ->  ran  { A }  e.  _V )
2624, 25syl 14 . . . . . . . . 9  |-  ( A  e.  _V  ->  ran  { A }  e.  _V )
27 uniexg 4422 . . . . . . . . 9  |-  ( ran 
{ A }  e.  _V  ->  U. ran  { A }  e.  _V )
2826, 27syl 14 . . . . . . . 8  |-  ( A  e.  _V  ->  U. ran  { A }  e.  _V )
29 opeq2 3764 . . . . . . . . . . 11  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
3029eqeq2d 2182 . . . . . . . . . 10  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
31 eleq1 2233 . . . . . . . . . . 11  |-  ( y  =  U. ran  { A }  ->  ( y  e.  C  <->  U. ran  { A }  e.  C
) )
3231anbi2d 461 . . . . . . . . . 10  |-  ( y  =  U. ran  { A }  ->  ( ( x  e.  B  /\  y  e.  C )  <->  ( x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
3330, 32anbi12d 470 . . . . . . . . 9  |-  ( 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 ) ) ) )
3433ceqsexgv 2859 . . . . . . . 8  |-  ( U. ran  { A }  e.  _V  ->  ( 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 )
) ) )
3528, 34syl 14 . . . . . . 7  |-  ( A  e.  _V  ->  ( 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 )
) ) )
3623, 35syl5bb 191 . . . . . 6  |-  ( A  e.  _V  ->  ( E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) )  <->  ( A  =  <. x ,  U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C )
) ) )
37 inteq 3832 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| A  =  |^| <. x ,  U. ran  { A } >. )
3837inteqd 3834 . . . . . . . . . . 11  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| |^| A  =  |^| |^|
<. x ,  U. ran  { A } >. )
3938adantl 275 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  |^| |^| A  =  |^| |^| <. x ,  U. ran  { A } >. )
40 op1stbg 4462 . . . . . . . . . . . 12  |-  ( ( x  e.  _V  /\  U.
ran  { A }  e.  _V )  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4115, 28, 40sylancr 412 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4241adantr 274 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4339, 42eqtr2d 2204 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  x  =  |^| |^| A )
4443ex 114 . . . . . . . 8  |-  ( A  e.  _V  ->  ( A  =  <. x , 
U. ran  { A } >.  ->  x  =  |^| |^| A ) )
4544pm4.71rd 392 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  =  <. x , 
U. ran  { A } >. 
<->  ( x  =  |^| |^| A  /\  A  = 
<. x ,  U. ran  { A } >. )
) )
4645anbi1d 462 . . . . . 6  |-  ( A  e.  _V  ->  (
( 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 ) ) ) )
47 anass 399 . . . . . . 7  |-  ( ( ( 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
) ) ) )
4847a1i 9 . . . . . 6  |-  ( A  e.  _V  ->  (
( ( 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
) ) ) ) )
4936, 46, 483bitrd 213 . . . . 5  |-  ( A  e.  _V  ->  ( 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
) ) ) ) )
5049exbidv 1818 . . . 4  |-  ( A  e.  _V  ->  ( 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 ) ) ) ) )
5111, 50syl5bb 191 . . 3  |-  ( A  e.  _V  ->  ( A  e.  ( B  X.  C )  <->  E. x
( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C )
) ) ) )
52 eqvisset 2740 . . . . . 6  |-  ( x  =  |^| |^| A  ->  |^| |^| A  e.  _V )
5352adantr 274 . . . . 5  |-  ( ( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )  ->  |^| |^| A  e.  _V )
5453exlimiv 1591 . . . 4  |-  ( E. x ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )  ->  |^| |^| A  e.  _V )
552ad2antrl 487 . . . 4  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  |^| |^| A  e.  _V )
56 opeq1 3763 . . . . . . 7  |-  ( x  =  |^| |^| A  -> 
<. x ,  U. ran  { A } >.  =  <. |^|
|^| A ,  U. ran  { A } >. )
5756eqeq2d 2182 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. |^|
|^| A ,  U. ran  { A } >. ) )
58 eleq1 2233 . . . . . . 7  |-  ( x  =  |^| |^| A  ->  ( x  e.  B  <->  |^|
|^| A  e.  B
) )
5958anbi1d 462 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( ( x  e.  B  /\  U. ran  { A }  e.  C
)  <->  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
6057, 59anbi12d 470 . . . . 5  |-  ( 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
) ) ) )
6160ceqsexgv 2859 . . . 4  |-  ( |^| |^| 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
) ) ) )
6254, 55, 61pm5.21nii 699 . . 3  |-  ( 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
) ) )
6351, 62bitrdi 195 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B  X.  C )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
641, 10, 63pm5.21nii 699 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:    /\ wa 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   {csn 3581   <.cop 3584   U.cuni 3794   |^|cint 3829    X. cxp 4607   ran crn 4610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-dm 4619  df-rn 4620
This theorem is referenced by: (None)
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