MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elxp5 Structured version   Unicode version

Theorem elxp5 5361
Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5360 when the double intersection does not create class existence problems (caused by int0 4066). (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 4898 . 2  |-  ( A  e.  ( B  X.  C )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) )
2 sneq 3827 . . . . . . . . . . . 12  |-  ( A  =  <. x ,  y
>.  ->  { A }  =  { <. x ,  y
>. } )
32rneqd 5100 . . . . . . . . . . 11  |-  ( A  =  <. x ,  y
>.  ->  ran  { A }  =  ran  { <. x ,  y >. } )
43unieqd 4028 . . . . . . . . . 10  |-  ( A  =  <. x ,  y
>.  ->  U. ran  { A }  =  U. ran  { <. x ,  y >. } )
5 vex 2961 . . . . . . . . . . 11  |-  x  e. 
_V
6 vex 2961 . . . . . . . . . . 11  |-  y  e. 
_V
75, 6op2nda 5357 . . . . . . . . . 10  |-  U. ran  {
<. x ,  y >. }  =  y
84, 7syl6req 2487 . . . . . . . . 9  |-  ( A  =  <. x ,  y
>.  ->  y  =  U. ran  { A } )
98pm4.71ri 616 . . . . . . . 8  |-  ( A  =  <. x ,  y
>. 
<->  ( y  =  U. ran  { A }  /\  A  =  <. x ,  y >. ) )
109anbi1i 678 . . . . . . 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 632 . . . . . . 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 242 . . . . . 6  |-  ( ( A  =  <. x ,  y >.  /\  (
x  e.  B  /\  y  e.  C )
)  <->  ( y  = 
U. ran  { A }  /\  ( A  = 
<. x ,  y >.  /\  ( x  e.  B  /\  y  e.  C
) ) ) )
1312exbii 1593 . . . . 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 4408 . . . . . . . 8  |-  { A }  e.  _V
1514rnex 5136 . . . . . . 7  |-  ran  { A }  e.  _V
1615uniex 4708 . . . . . 6  |-  U. ran  { A }  e.  _V
17 opeq2 3987 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  <. x ,  y >.  =  <. x ,  U. ran  { A } >. )
1817eqeq2d 2449 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( A  =  <. x ,  y
>. 
<->  A  =  <. x ,  U. ran  { A } >. ) )
19 eleq1 2498 . . . . . . . 8  |-  ( y  =  U. ran  { A }  ->  ( y  e.  C  <->  U. ran  { A }  e.  C
) )
2019anbi2d 686 . . . . . . 7  |-  ( y  =  U. ran  { A }  ->  ( ( x  e.  B  /\  y  e.  C )  <->  ( x  e.  B  /\  U.
ran  { A }  e.  C ) ) )
2118, 20anbi12d 693 . . . . . 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 2993 . . . . 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 242 . . . 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 4055 . . . . . . . 8  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| A  =  |^| <. x ,  U. ran  { A } >. )
2524inteqd 4057 . . . . . . 7  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  |^| |^| A  =  |^| |^|
<. x ,  U. ran  { A } >. )
265, 16op1stb 4761 . . . . . . 7  |-  |^| |^| <. x ,  U. ran  { A } >.  =  x
2725, 26syl6req 2487 . . . . . 6  |-  ( A  =  <. x ,  U. ran  { A } >.  ->  x  =  |^| |^| A
)
2827pm4.71ri 616 . . . . 5  |-  ( A  =  <. x ,  U. ran  { A } >.  <->  (
x  =  |^| |^| A  /\  A  =  <. x ,  U. ran  { A } >. ) )
2928anbi1i 678 . . . 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 632 . . . 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 264 . . 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 1593 . 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 eleq1 2498 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( x  e.  _V  <->  |^|
|^| A  e.  _V ) )
345, 33mpbii 204 . . . . 5  |-  ( x  =  |^| |^| A  ->  |^| |^| A  e.  _V )
3534adantr 453 . . . 4  |-  ( ( x  =  |^| |^| A  /\  ( A  =  <. x ,  U. ran  { A } >.  /\  (
x  e.  B  /\  U.
ran  { A }  e.  C ) ) )  ->  |^| |^| A  e.  _V )
3635exlimiv 1645 . . 3  |-  ( E. x ( x  = 
|^| |^| A  /\  ( A  =  <. x , 
U. ran  { A } >.  /\  ( x  e.  B  /\  U. ran  { A }  e.  C
) ) )  ->  |^| |^| A  e.  _V )
37 elex 2966 . . . 4  |-  ( |^| |^| A  e.  B  ->  |^| |^| A  e.  _V )
3837ad2antrl 710 . . 3  |-  ( ( A  =  <. |^| |^| A ,  U. ran  { A } >.  /\  ( |^| |^| A  e.  B  /\  U.
ran  { A }  e.  C ) )  ->  |^| |^| A  e.  _V )
39 opeq1 3986 . . . . . 6  |-  ( x  =  |^| |^| A  -> 
<. x ,  U. ran  { A } >.  =  <. |^|
|^| A ,  U. ran  { A } >. )
4039eqeq2d 2449 . . . . 5  |-  ( x  =  |^| |^| A  ->  ( A  =  <. x ,  U. ran  { A } >.  <->  A  =  <. |^|
|^| A ,  U. ran  { A } >. ) )
41 eleq1 2498 . . . . . 6  |-  ( x  =  |^| |^| A  ->  ( x  e.  B  <->  |^|
|^| A  e.  B
) )
4241anbi1d 687 . . . . 5  |-  ( x  =  |^| |^| A  ->  ( ( x  e.  B  /\  U. ran  { A }  e.  C
)  <->  ( |^| |^| A  e.  B  /\  U. ran  { A }  e.  C
) ) )
4340, 42anbi12d 693 . . . 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
) ) ) )
4443ceqsexgv 3070 . . 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
) ) ) )
4536, 38, 44pm5.21nii 344 . 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
) ) )
461, 32, 453bitri 264 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:    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   <.cop 3819   U.cuni 4017   |^|cint 4052    X. cxp 4879   ran crn 4882
This theorem is referenced by:  xpnnenOLD  12814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-dm 4891  df-rn 4892
  Copyright terms: Public domain W3C validator