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Theorem elxp5 4839
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4838 when the double intersection does not create class existence problems (caused by int0 3658). (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 2611 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 elex 2611 . . . 4  |-  ( |^| |^| A  e.  B  ->  |^| |^| A  e.  _V )
3 elex 2611 . . . 4  |-  ( U. ran  { A }  e.  C  ->  U. ran  { A }  e.  _V )
42, 3anim12i 331 . . 3  |-  ( (
|^| |^| A  e.  B  /\  U. ran  { A }  e.  C )  ->  ( |^| |^| A  e.  _V  /\  U. ran  { A }  e.  _V ) )
5 opexg 3991 . . . . 5  |-  ( (
|^| |^| A  e.  _V  /\ 
U. ran  { A }  e.  _V )  -> 
<. |^| |^| A ,  U. ran  { A } >.  e. 
_V )
65adantl 271 . . . 4  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  _V  /\  U.
ran  { A }  e.  _V ) )  ->  <. |^| |^| A ,  U. ran  { A } >.  e.  _V )
7 eleq1 2142 . . . . 5  |-  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  -> 
( A  e.  _V  <->  <. |^| |^| A ,  U. ran  { A } >.  e. 
_V ) )
87adantr 270 . . . 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 165 . . 3  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  _V  /\  U.
ran  { A }  e.  _V ) )  ->  A  e.  _V )
104, 9sylan2 280 . 2  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  A  e.  _V )
11 elxp 4388 . . . 4  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
12 sneq 3417 . . . . . . . . . . . . . 14  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
1312rneqd 4591 . . . . . . . . . . . . 13  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
1413unieqd 3620 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
15 vex 2605 . . . . . . . . . . . . 13  |-  x  e. 
_V
16 vex 2605 . . . . . . . . . . . . 13  |-  y  e. 
_V
1715, 16op2nda 4835 . . . . . . . . . . . 12  |-  U. ran  {
<. x ,  y >. }  =  y
1814, 17syl6req 2131 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  y  =  U. ran  { A } )
1918pm4.71ri 384 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>. 
<->  ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. ) )
2019anbi1i 446 . . . . . . . . 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 393 . . . . . . . . 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 182 . . . . . . . 8  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
2322exbii 1537 . . . . . . 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 3964 . . . . . . . . . 10  |-  ( A  e.  _V  ->  { A }  e.  _V )
25 rnexg 4625 . . . . . . . . . 10  |-  ( { A }  e.  _V  ->  ran  { A }  e.  _V )
2624, 25syl 14 . . . . . . . . 9  |-  ( A  e.  _V  ->  ran  { A }  e.  _V )
27 uniexg 4201 . . . . . . . . 9  |-  ( ran 
{ A }  e.  _V  ->  U. ran  { A }  e.  _V )
2826, 27syl 14 . . . . . . . 8  |-  ( A  e.  _V  ->  U. ran  { A }  e.  _V )
29 opeq2 3579 . . . . . . . . . . 11  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
3029eqeq2d 2093 . . . . . . . . . 10  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
31 eleq1 2142 . . . . . . . . . . 11  |-  ( y  =  U. ran  { A }  ->  ( y  e.  C  <->  U. ran  { A }  e.  C
) )
3231anbi2d 452 . . . . . . . . . 10  |-  ( y  =  U. ran  { A }  ->  ( ( x  e.  B  /\  y  e.  C )  <->  ( x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
3330, 32anbi12d 457 . . . . . . . . 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 2725 . . . . . . . 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 190 . . . . . 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 3647 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| A  =  |^| <. x ,  U. ran  { A } >. )
3837inteqd 3649 . . . . . . . . . . 11  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| |^| A  =  |^| |^|
<. x ,  U. ran  { A } >. )
3938adantl 271 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  |^| |^| A  =  |^| |^| <. x ,  U. ran  { A } >. )
40 op1stbg 4236 . . . . . . . . . . . 12  |-  ( ( x  e.  _V  /\  U.
ran  { A }  e.  _V )  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4115, 28, 40sylancr 405 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4241adantr 270 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4339, 42eqtr2d 2115 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  x  =  |^| |^| A )
4443ex 113 . . . . . . . 8  |-  ( A  e.  _V  ->  ( A  =  <. x , 
U. ran  { A } >.  ->  x  =  |^| |^| A ) )
4544pm4.71rd 386 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  =  <. x , 
U. ran  { A } >. 
<->  ( x  =  |^| |^| A  /\  A  = 
<. x ,  U. ran  { A } >. )
) )
4645anbi1d 453 . . . . . 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 393 . . . . . . 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 212 . . . . 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 1747 . . . 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 190 . . 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 2610 . . . . . 6  |-  ( x  =  |^| |^| A  ->  |^| |^| A  e.  _V )
5352adantr 270 . . . . 5  |-  ( ( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )  ->  |^| |^| A  e.  _V )
5453exlimiv 1530 . . . 4  |-  ( E. x ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )  ->  |^| |^| A  e.  _V )
552ad2antrl 474 . . . 4  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  |^| |^| A  e.  _V )
56 opeq1 3578 . . . . . . 7  |-  ( x  =  |^| |^| A  -> 
<. x ,  U. ran  { A } >.  =  <. |^|
|^| A ,  U. ran  { A } >. )
5756eqeq2d 2093 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. |^|
|^| A ,  U. ran  { A } >. ) )
58 eleq1 2142 . . . . . . 7  |-  ( x  =  |^| |^| A  ->  ( x  e.  B  <->  |^|
|^| A  e.  B
) )
5958anbi1d 453 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( ( x  e.  B  /\  U. ran  { A }  e.  C
)  <->  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
6057, 59anbi12d 457 . . . . 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 2725 . . . 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 653 . . 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, 62syl6bb 194 . 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 653 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 102    <-> wb 103    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2602   {csn 3406   <.cop 3409   U.cuni 3609   |^|cint 3644    X. cxp 4369   ran crn 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382
This theorem is referenced by: (None)
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