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Theorem elxp6 6318
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5298. (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 elxp4 5298 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. U. dom  { A } ,  U. ran  { A } >.  /\  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) ) )
2 1stval 6291 . . . . 5  |-  ( 1st `  A )  =  U. dom  { A }
3 2ndval 6292 . . . . 5  |-  ( 2nd `  A )  =  U. ran  { A }
42, 3opeq12i 3932 . . . 4  |-  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >.
54eqeq2i 2398 . . 3  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  <->  A  =  <. U. dom  { A } ,  U. ran  { A } >. )
62eleq1i 2451 . . . 4  |-  ( ( 1st `  A )  e.  B  <->  U. dom  { A }  e.  B
)
73eleq1i 2451 . . . 4  |-  ( ( 2nd `  A )  e.  C  <->  U. ran  { A }  e.  C
)
86, 7anbi12i 679 . . 3  |-  ( ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) )
95, 8anbi12i 679 . 2  |-  ( ( 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
) ) )
101, 9bitr4i 244 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:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3758   <.cop 3761   U.cuni 3958    X. cxp 4817   dom cdm 4819   ran crn 4820   ` cfv 5395   1stc1st 6287   2ndc2nd 6288
This theorem is referenced by:  elxp7  6319  eqopi  6323  1st2nd2  6326  r0weon  7828  qredeu  13035  qnumdencl  13059  tx1cn  17563  tx2cn  17564  txhaus  17601  xppreima  23902  1stmbfm  24405  2ndmbfm  24406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fv 5403  df-1st 6289  df-2nd 6290
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