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Theorem elxp6 6167
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5176. (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 5176 . 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 6140 . . . . 5  |-  ( 1st `  A )  =  U. dom  { A }
3 2ndval 6141 . . . . 5  |-  ( 2nd `  A )  =  U. ran  { A }
42, 3opeq12i 3817 . . . 4  |-  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. U. dom  { A } ,  U. ran  { A } >.
54eqeq2i 2306 . . 3  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  <->  A  =  <. U. dom  { A } ,  U. ran  { A } >. )
62eleq1i 2359 . . . 4  |-  ( ( 1st `  A )  e.  B  <->  U. dom  { A }  e.  B
)
73eleq1i 2359 . . . 4  |-  ( ( 2nd `  A )  e.  C  <->  U. ran  { A }  e.  C
)
86, 7anbi12i 678 . . 3  |-  ( ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C )  <->  ( U. dom  { A }  e.  B  /\  U. ran  { A }  e.  C
) )
95, 8anbi12i 678 . 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 243 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   {csn 3653   <.cop 3656   U.cuni 3843    X. cxp 4703   dom cdm 4705   ran crn 4706   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  elxp7  6168  eqopi  6172  1st2nd2  6175  r0weon  7656  qredeu  12802  qnumdencl  12826  tx1cn  17319  tx2cn  17320  txhaus  17357  xppreima  23226  1stmbfm  23580  2ndmbfm  23581  prj1b  25182  prj3  25183  issubcata  25949  fnctartar  26010  fnctartar2  26011
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-1st 6138  df-2nd 6139
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