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Theorem elxp6 5954
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4931. (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 2631 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 opexg 4064 . . . 4  |-  ( ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _V )
32adantl 272 . . 3  |-  ( ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  _V )
4 eleq1 2151 . . . 4  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  ( A  e.  _V  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  _V ) )
54adantr 271 . . 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 5927 . . . . . 6  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
8 2ndvalg 5928 . . . . . 6  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
97, 8opeq12d 3636 . . . . 5  |-  ( A  e.  _V  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >. )
109eqeq2d 2100 . . . 4  |-  ( A  e.  _V  ->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. 
<->  A  =  <. U. dom  { A } ,  U. ran  { A } >. ) )
117eleq1d 2157 . . . . 5  |-  ( A  e.  _V  ->  (
( 1st `  A
)  e.  B  <->  U. dom  { A }  e.  B
) )
128eleq1d 2157 . . . . 5  |-  ( A  e.  _V  ->  (
( 2nd `  A
)  e.  C  <->  U. ran  { A }  e.  C
) )
1311, 12anbi12d 458 . . . 4  |-  ( A  e.  _V  ->  (
( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
1410, 13anbi12d 458 . . 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 4931 . . 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 656 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 1290    e. wcel 1439   _Vcvv 2620   {csn 3450   <.cop 3453   U.cuni 3659    X. cxp 4450   dom cdm 4452   ran crn 4453   ` cfv 5028   1stc1st 5923   2ndc2nd 5924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-iota 4993  df-fun 5030  df-fv 5036  df-1st 5925  df-2nd 5926
This theorem is referenced by:  elxp7  5955  eqopi  5956  1st2nd2  5959  eldju2ndl  6817  eldju2ndr  6818  qredeu  11418  qnumdencl  11504
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