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Theorem elxp6 6607
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 6521. (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 6521 . 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 6578 . . . . 5  |-  ( 1st `  A )  =  U. dom  { A }
3 2ndval 6579 . . . . 5  |-  ( 2nd `  A )  =  U. ran  { A }
42, 3opeq12i 4063 . . . 4  |-  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >.
54eqeq2i 2452 . . 3  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  <->  A  =  <. U. dom  { A } ,  U. ran  { A } >. )
62eleq1i 2505 . . . 4  |-  ( ( 1st `  A )  e.  B  <->  U. dom  { A }  e.  B
)
73eleq1i 2505 . . . 4  |-  ( ( 2nd `  A )  e.  C  <->  U. ran  { A }  e.  C
)
86, 7anbi12i 697 . . 3  |-  ( ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) )
95, 8anbi12i 697 . 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 252 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 184    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3876   <.cop 3882   U.cuni 4090    X. cxp 4837   dom cdm 4839   ran crn 4840   ` cfv 5417   1stc1st 6574   2ndc2nd 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fv 5425  df-1st 6576  df-2nd 6577
This theorem is referenced by:  elxp7  6608  eqopi  6609  1st2nd2  6612  r0weon  8178  qredeu  13792  qnumdencl  13816  tx1cn  19181  tx2cn  19182  txhaus  19219  psmetxrge0  19888  xppreima  25963  ofpreima2  25984  1stmbfm  26674  2ndmbfm  26675  oddpwdcv  26737
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