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Theorem elxp6 6118
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5074. (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 2723 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 opexg 4189 . . . 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 2220 . . . 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 elxp4 5074 . . 3  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
8 1stvalg 6091 . . . . . 6  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
9 2ndvalg 6092 . . . . . 6  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
108, 9opeq12d 3750 . . . . 5  |-  ( A  e.  _V  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >. )
1110eqeq2d 2169 . . . 4  |-  ( A  e.  _V  ->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. 
<->  A  =  <. U. dom  { A } ,  U. ran  { A } >. ) )
128eleq1d 2226 . . . . 5  |-  ( A  e.  _V  ->  (
( 1st `  A
)  e.  B  <->  U. dom  { A }  e.  B
) )
139eleq1d 2226 . . . . 5  |-  ( A  e.  _V  ->  (
( 2nd `  A
)  e.  C  <->  U. ran  { A }  e.  C
) )
1412, 13anbi12d 465 . . . 4  |-  ( A  e.  _V  ->  (
( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
1511, 14anbi12d 465 . . 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
) ) ) )
167, 15bitr4id 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 694 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 1335    e. wcel 2128   _Vcvv 2712   {csn 3560   <.cop 3563   U.cuni 3773    X. cxp 4585   dom cdm 4587   ran crn 4588   ` cfv 5171   1stc1st 6087   2ndc2nd 6088
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-iota 5136  df-fun 5173  df-fv 5179  df-1st 6089  df-2nd 6090
This theorem is referenced by:  elxp7  6119  oprssdmm  6120  eqopi  6121  1st2nd2  6124  eldju2ndl  7017  eldju2ndr  7018  qredeu  11978  qnumdencl  12066  tx1cn  12711  tx2cn  12712  psmetxrge0  12774  xmetxpbl  12950
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