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Theorem elxp6 6035
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4996. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
elxp6  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )

Proof of Theorem elxp6
StepHypRef Expression
1 elex 2671 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 opexg 4120 . . . 4  |-  ( ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _V )
32adantl 275 . . 3  |-  ( ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  _V )
4 eleq1 2180 . . . 4  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  ( A  e.  _V  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  _V ) )
54adantr 274 . . 3  |-  ( ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) )  ->  ( A  e.  _V  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  _V ) )
63, 5mpbird 166 . 2  |-  ( ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) )  ->  A  e.  _V )
7 1stvalg 6008 . . . . . 6  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
8 2ndvalg 6009 . . . . . 6  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
97, 8opeq12d 3683 . . . . 5  |-  ( A  e.  _V  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >. )
109eqeq2d 2129 . . . 4  |-  ( A  e.  _V  ->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. 
<->  A  =  <. U. dom  { A } ,  U. ran  { A } >. ) )
117eleq1d 2186 . . . . 5  |-  ( A  e.  _V  ->  (
( 1st `  A
)  e.  B  <->  U. dom  { A }  e.  B
) )
128eleq1d 2186 . . . . 5  |-  ( A  e.  _V  ->  (
( 2nd `  A
)  e.  C  <->  U. ran  { A }  e.  C
) )
1311, 12anbi12d 464 . . . 4  |-  ( A  e.  _V  ->  (
( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
1410, 13anbi12d 464 . . 3  |-  ( A  e.  _V  ->  (
( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) ) )
15 elxp4 4996 . . 3  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
1614, 15syl6rbbr 198 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) ) )
171, 6, 16pm5.21nii 678 1  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   _Vcvv 2660   {csn 3497   <.cop 3500   U.cuni 3706    X. cxp 4507   dom cdm 4509   ran crn 4510   ` cfv 5093   1stc1st 6004   2ndc2nd 6005
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fv 5101  df-1st 6006  df-2nd 6007
This theorem is referenced by:  elxp7  6036  oprssdmm  6037  eqopi  6038  1st2nd2  6041  eldju2ndl  6925  eldju2ndr  6926  qredeu  11705  qnumdencl  11792  tx1cn  12365  tx2cn  12366  psmetxrge0  12428  xmetxpbl  12604
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