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Theorem elxp5 4787
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 4786 when the double intersection does not create class existence problems (caused by int0 3626). (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 2563 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 elex 2563 . . . 4  |-  ( |^| |^| A  e.  B  ->  |^| |^| A  e.  _V )
3 elex 2563 . . . 4  |-  ( U. ran  { A }  e.  C  ->  U. ran  { A }  e.  _V )
42, 3anim12i 321 . . 3  |-  ( (
|^| |^| A  e.  B  /\  U. ran  { A }  e.  C )  ->  ( |^| |^| A  e.  _V  /\  U. ran  { A }  e.  _V ) )
5 opexgOLD 3962 . . . . 5  |-  ( (
|^| |^| A  e.  _V  /\ 
U. ran  { A }  e.  _V )  -> 
<. |^| |^| A ,  U. ran  { A } >.  e. 
_V )
65adantl 262 . . . 4  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  _V  /\  U.
ran  { A }  e.  _V ) )  ->  <. |^| |^| A ,  U. ran  { A } >.  e.  _V )
7 eleq1 2100 . . . . 5  |-  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  -> 
( A  e.  _V  <->  <. |^| |^| A ,  U. ran  { A } >.  e. 
_V ) )
87adantr 261 . . . 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 156 . . 3  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  _V  /\  U.
ran  { A }  e.  _V ) )  ->  A  e.  _V )
104, 9sylan2 270 . 2  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  A  e.  _V )
11 elxp 4340 . . . 4  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
12 sneq 3383 . . . . . . . . . . . . . 14  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
1312rneqd 4541 . . . . . . . . . . . . 13  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
1413unieqd 3588 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
15 vex 2557 . . . . . . . . . . . . 13  |-  x  e. 
_V
16 vex 2557 . . . . . . . . . . . . 13  |-  y  e. 
_V
1715, 16op2nda 4783 . . . . . . . . . . . 12  |-  U. ran  {
<. x ,  y >. }  =  y
1814, 17syl6req 2089 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  y  =  U. ran  { A } )
1918pm4.71ri 372 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>. 
<->  ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. ) )
2019anbi1i 431 . . . . . . . . 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 381 . . . . . . . . 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 173 . . . . . . . 8  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
2322exbii 1496 . . . . . . 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 snexgOLD 3932 . . . . . . . . . 10  |-  ( A  e.  _V  ->  { A }  e.  _V )
25 rnexg 4575 . . . . . . . . . 10  |-  ( { A }  e.  _V  ->  ran  { A }  e.  _V )
2624, 25syl 14 . . . . . . . . 9  |-  ( A  e.  _V  ->  ran  { A }  e.  _V )
27 uniexg 4162 . . . . . . . . 9  |-  ( ran 
{ A }  e.  _V  ->  U. ran  { A }  e.  _V )
2826, 27syl 14 . . . . . . . 8  |-  ( A  e.  _V  ->  U. ran  { A }  e.  _V )
29 opeq2 3547 . . . . . . . . . . 11  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
3029eqeq2d 2051 . . . . . . . . . 10  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
31 eleq1 2100 . . . . . . . . . . 11  |-  ( y  =  U. ran  { A }  ->  ( y  e.  C  <->  U. ran  { A }  e.  C
) )
3231anbi2d 437 . . . . . . . . . 10  |-  ( y  =  U. ran  { A }  ->  ( ( x  e.  B  /\  y  e.  C )  <->  ( x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
3330, 32anbi12d 442 . . . . . . . . 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 2670 . . . . . . . 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 181 . . . . . 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 3615 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| A  =  |^| <. x ,  U. ran  { A } >. )
3837inteqd 3617 . . . . . . . . . . 11  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| |^| A  =  |^| |^|
<. x ,  U. ran  { A } >. )
3938adantl 262 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  |^| |^| A  =  |^| |^| <. x ,  U. ran  { A } >. )
40 op1stbg 4197 . . . . . . . . . . . 12  |-  ( ( x  e.  _V  /\  U.
ran  { A }  e.  _V )  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4115, 28, 40sylancr 393 . . . . . . . . . . 11  |-  ( A  e.  _V  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4241adantr 261 . . . . . . . . . 10  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  |^| |^| <. x ,  U. ran  { A } >.  =  x )
4339, 42eqtr2d 2073 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  A  =  <. x , 
U. ran  { A } >. )  ->  x  =  |^| |^| A )
4443ex 108 . . . . . . . 8  |-  ( A  e.  _V  ->  ( A  =  <. x , 
U. ran  { A } >.  ->  x  =  |^| |^| A ) )
4544pm4.71rd 374 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  =  <. x , 
U. ran  { A } >. 
<->  ( x  =  |^| |^| A  /\  A  = 
<. x ,  U. ran  { A } >. )
) )
4645anbi1d 438 . . . . . 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 381 . . . . . . 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 203 . . . . 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 1706 . . . 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 181 . . 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 eleq1 2100 . . . . . . 7  |-  ( x  =  |^| |^| A  ->  ( x  e.  _V  <->  |^|
|^| A  e.  _V ) )
5315, 52mpbii 136 . . . . . 6  |-  ( x  =  |^| |^| A  ->  |^| |^| A  e.  _V )
5453adantr 261 . . . . 5  |-  ( ( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )  ->  |^| |^| A  e.  _V )
5554exlimiv 1489 . . . 4  |-  ( E. x ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )  ->  |^| |^| A  e.  _V )
562ad2antrl 459 . . . 4  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  |^| |^| A  e.  _V )
57 opeq1 3546 . . . . . . 7  |-  ( x  =  |^| |^| A  -> 
<. x ,  U. ran  { A } >.  =  <. |^|
|^| A ,  U. ran  { A } >. )
5857eqeq2d 2051 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. |^|
|^| A ,  U. ran  { A } >. ) )
59 eleq1 2100 . . . . . . 7  |-  ( x  =  |^| |^| A  ->  ( x  e.  B  <->  |^|
|^| A  e.  B
) )
6059anbi1d 438 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( ( x  e.  B  /\  U. ran  { A }  e.  C
)  <->  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
6158, 60anbi12d 442 . . . . 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
) ) ) )
6261ceqsexgv 2670 . . . 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
) ) ) )
6355, 56, 62pm5.21nii 620 . . 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
) ) )
6451, 63syl6bb 185 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B  X.  C )  <->  ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
651, 10, 64pm5.21nii 620 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 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   _Vcvv 2554   {csn 3372   <.cop 3375   U.cuni 3577   |^|cint 3612    X. cxp 4321   ran crn 4324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4157
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-br 3762  df-opab 3816  df-xp 4329  df-rel 4330  df-cnv 4331  df-dm 4333  df-rn 4334
This theorem is referenced by: (None)
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