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Theorem elxp5 5027
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5026 when the double intersection does not create class existence problems (caused by int0 3785). (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 2697 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 elex 2697 . . . 4  |-  ( |^| |^| A  e.  B  ->  |^| |^| A  e.  _V )
3 elex 2697 . . . 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 4150 . . . . 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 2202 . . . . 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 4556 . . . 4  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
12 sneq 3538 . . . . . . . . . . . . . 14  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
1312rneqd 4768 . . . . . . . . . . . . 13  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
1413unieqd 3747 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
15 vex 2689 . . . . . . . . . . . . 13  |-  x  e. 
_V
16 vex 2689 . . . . . . . . . . . . 13  |-  y  e. 
_V
1715, 16op2nda 5023 . . . . . . . . . . . 12  |-  U. ran  {
<. x ,  y >. }  =  y
1814, 17syl6req 2189 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  y  =  U. ran  { A } )
1918pm4.71ri 389 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>. 
<->  ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. ) )
2019anbi1i 453 . . . . . . . . 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 398 . . . . . . . . 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 1584 . . . . . . 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 4108 . . . . . . . . . 10  |-  ( A  e.  _V  ->  { A }  e.  _V )
25 rnexg 4804 . . . . . . . . . 10  |-  ( { A }  e.  _V  ->  ran  { A }  e.  _V )
2624, 25syl 14 . . . . . . . . 9  |-  ( A  e.  _V  ->  ran  { A }  e.  _V )
27 uniexg 4361 . . . . . . . . 9  |-  ( ran 
{ A }  e.  _V  ->  U. ran  { A }  e.  _V )
2826, 27syl 14 . . . . . . . 8  |-  ( A  e.  _V  ->  U. ran  { A }  e.  _V )
29 opeq2 3706 . . . . . . . . . . 11  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
3029eqeq2d 2151 . . . . . . . . . 10  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
31 eleq1 2202 . . . . . . . . . . 11  |-  ( y  =  U. ran  { A }  ->  ( y  e.  C  <->  U. ran  { A }  e.  C
) )
3231anbi2d 459 . . . . . . . . . 10  |-  ( y  =  U. ran  { A }  ->  ( ( x  e.  B  /\  y  e.  C )  <->  ( x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
3330, 32anbi12d 464 . . . . . . . . 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 2814 . . . . . . . 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 3774 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| A  =  |^| <. x ,  U. ran  { A } >. )
3837inteqd 3776 . . . . . . . . . . 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 4400 . . . . . . . . . . . 12  |-  ( ( x  e.  _V  /\  U.
ran  { A }  e.  _V )  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4115, 28, 40sylancr 410 . . . . . . . . . . 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 2173 . . . . . . . . 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 391 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  =  <. x , 
U. ran  { A } >. 
<->  ( x  =  |^| |^| A  /\  A  = 
<. x ,  U. ran  { A } >. )
) )
4645anbi1d 460 . . . . . 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 398 . . . . . . 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 1797 . . . 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 2696 . . . . . 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 1577 . . . 4  |-  ( E. x ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )  ->  |^| |^| A  e.  _V )
552ad2antrl 481 . . . 4  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  |^| |^| A  e.  _V )
56 opeq1 3705 . . . . . . 7  |-  ( x  =  |^| |^| A  -> 
<. x ,  U. ran  { A } >.  =  <. |^|
|^| A ,  U. ran  { A } >. )
5756eqeq2d 2151 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. |^|
|^| A ,  U. ran  { A } >. ) )
58 eleq1 2202 . . . . . . 7  |-  ( x  =  |^| |^| A  ->  ( x  e.  B  <->  |^|
|^| A  e.  B
) )
5958anbi1d 460 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( ( x  e.  B  /\  U. ran  { A }  e.  C
)  <->  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
6057, 59anbi12d 464 . . . . 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 2814 . . . 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 693 . . 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 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 693 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 1331   E.wex 1468    e. wcel 1480   _Vcvv 2686   {csn 3527   <.cop 3530   U.cuni 3736   |^|cint 3771    X. cxp 4537   ran crn 4540
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550
This theorem is referenced by: (None)
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