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Theorem elxp6 5824
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4836. (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 2583 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 opexg 3992 . . . 4  |-  ( ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  ->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  _V )
32adantl 266 . . 3  |-  ( ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  _V )
4 eleq1 2116 . . . 4  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  ( A  e.  _V  <->  <. ( 1st `  A ) ,  ( 2nd `  A )
>.  e.  _V ) )
54adantr 265 . . 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 160 . 2  |-  ( ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) )  ->  A  e.  _V )
7 1stvalg 5797 . . . . . 6  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
8 2ndvalg 5798 . . . . . 6  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
97, 8opeq12d 3585 . . . . 5  |-  ( A  e.  _V  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >. )
109eqeq2d 2067 . . . 4  |-  ( A  e.  _V  ->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. 
<->  A  =  <. U. dom  { A } ,  U. ran  { A } >. ) )
117eleq1d 2122 . . . . 5  |-  ( A  e.  _V  ->  (
( 1st `  A
)  e.  B  <->  U. dom  { A }  e.  B
) )
128eleq1d 2122 . . . . 5  |-  ( A  e.  _V  ->  (
( 2nd `  A
)  e.  C  <->  U. ran  { A }  e.  C
) )
1311, 12anbi12d 450 . . . 4  |-  ( A  e.  _V  ->  (
( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
1410, 13anbi12d 450 . . 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 4836 . . 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 192 . 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 630 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 101    <-> wb 102    = wceq 1259    e. wcel 1409   _Vcvv 2574   {csn 3403   <.cop 3406   U.cuni 3608    X. cxp 4371   dom cdm 4373   ran crn 4374   ` cfv 4930   1stc1st 5793   2ndc2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by:  elxp7  5825  eqopi  5826  1st2nd2  5829
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