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Theorem elxp6 6731
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 6643. (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 6643 . 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 6701 . . . . 5  |-  ( 1st `  A )  =  U. dom  { A }
3 2ndval 6702 . . . . 5  |-  ( 2nd `  A )  =  U. ran  { A }
42, 3opeq12i 4136 . . . 4  |-  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >.
54eqeq2i 2400 . . 3  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  <->  A  =  <. U. dom  { A } ,  U. ran  { A } >. )
62eleq1i 2459 . . . 4  |-  ( ( 1st `  A )  e.  B  <->  U. dom  { A }  e.  B
)
73eleq1i 2459 . . . 4  |-  ( ( 2nd `  A )  e.  C  <->  U. ran  { A }  e.  C
)
86, 7anbi12i 695 . . 3  |-  ( ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) )
95, 8anbi12i 695 . 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 367    = wceq 1399    e. wcel 1826   {csn 3944   <.cop 3950   U.cuni 4163    X. cxp 4911   dom cdm 4913   ran crn 4914   ` cfv 5496   1stc1st 6697   2ndc2nd 6698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fv 5504  df-1st 6699  df-2nd 6700
This theorem is referenced by:  elxp7  6732  eqopi  6733  1st2nd2  6736  r0weon  8303  qredeu  14250  qnumdencl  14274  tx1cn  20195  tx2cn  20196  txhaus  20233  psmetxrge0  20902  xppreima  27627  ofpreima2  27654  1stmbfm  28387  2ndmbfm  28388  oddpwdcv  28477
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