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Theorem 1st2nd 6352
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 4844 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3303 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 459 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6345 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
53, 4syl 16 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    C_ wss 3280   <.cop 3777    X. cxp 4835   Rel wrel 4842   ` cfv 5413   1stc1st 6306   2ndc2nd 6307
This theorem is referenced by:  2ndrn  6354  1st2ndbr  6355  elopabi  6371  cnvf1olem  6403  ordpinq  8776  addassnq  8791  mulassnq  8792  distrnq  8794  mulidnq  8796  recmulnq  8797  ltexnq  8808  fsumcnv  12512  cofulid  14042  cofurid  14043  idffth  14085  cofull  14086  cofth  14087  ressffth  14090  isnat2  14100  nat1st2nd  14103  homadmcd  14152  catciso  14217  prf1st  14256  prf2nd  14257  1st2ndprf  14258  curfuncf  14290  uncfcurf  14291  curf2ndf  14299  yonffthlem  14334  yoniso  14337  dprd2dlem2  15553  dprd2dlem1  15554  dprd2da  15555  2ndcctbss  17471  utop2nei  18233  utop3cls  18234  caubl  19213  rngoi  21921  drngoi  21948  nvop2  22040  nvvop  22041  nvop  22119  phop  22272  cvmliftlem1  24925  fprodcnv  25260  heiborlem3  26412  isdrngo1  26462  iscrngo2  26498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fv 5421  df-1st 6308  df-2nd 6309
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