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Theorem 1st2nd 6166
Description: Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.)
Assertion
Ref Expression
1st2nd  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )

Proof of Theorem 1st2nd
StepHypRef Expression
1 df-rel 4696 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3175 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 458 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6159 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
53, 4syl 15 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   <.cop 3643    X. cxp 4687   Rel wrel 4694   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  2ndrn  6168  1st2ndbr  6169  elopabi  6185  cnvf1olem  6216  ordpinq  8567  addassnq  8582  mulassnq  8583  distrnq  8585  mulidnq  8587  recmulnq  8588  ltexnq  8599  fsumcnv  12236  cofulid  13764  cofurid  13765  idffth  13807  cofull  13808  cofth  13809  ressffth  13812  isnat2  13822  nat1st2nd  13825  homadmcd  13874  catciso  13939  prf1st  13978  prf2nd  13979  1st2ndprf  13980  curfuncf  14012  uncfcurf  14013  curf2ndf  14021  yonffthlem  14056  yoniso  14059  dprd2dlem2  15275  dprd2dlem1  15276  dprd2da  15277  2ndcctbss  17181  caubl  18733  rngoi  21047  drngoi  21074  nvop2  21164  nvvop  21165  nvop  21243  phop  21396  cvmliftlem1  23816  11st22nd  25045  heiborlem3  26537  isdrngo1  26587  iscrngo2  26623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-1st 6122  df-2nd 6123
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