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Theorem elxp6 6852
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 6764. (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 6764 . 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 6822 . . . . 5  |-  ( 1st `  A )  =  U. dom  { A }
3 2ndval 6823 . . . . 5  |-  ( 2nd `  A )  =  U. ran  { A }
42, 3opeq12i 4185 . . . 4  |-  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >.
54eqeq2i 2474 . . 3  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  <->  A  =  <. U. dom  { A } ,  U. ran  { A } >. )
62eleq1i 2531 . . . 4  |-  ( ( 1st `  A )  e.  B  <->  U. dom  { A }  e.  B
)
73eleq1i 2531 . . . 4  |-  ( ( 2nd `  A )  e.  C  <->  U. ran  { A }  e.  C
)
86, 7anbi12i 708 . . 3  |-  ( ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) )
95, 8anbi12i 708 . 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 260 1  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   {csn 3980   <.cop 3986   U.cuni 4212    X. cxp 4851   dom cdm 4853   ran crn 4854   ` cfv 5601   1stc1st 6818   2ndc2nd 6819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-1st 6820  df-2nd 6821
This theorem is referenced by:  elxp7  6853  eqopi  6854  1st2nd2  6857  r0weon  8469  qredeu  14713  qnumdencl  14737  tx1cn  20673  tx2cn  20674  txhaus  20711  psmetxrge0  21378  xppreima  28297  ofpreima2  28318  smatrcl  28671  1stmbfm  29131  2ndmbfm  29132  oddpwdcv  29237
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