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Theorem 1st2nd 6293
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 4799 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3261 . . 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 6286 . 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 1647    e. wcel 1715   _Vcvv 2873    C_ wss 3238   <.cop 3732    X. cxp 4790   Rel wrel 4797   ` cfv 5358   1stc1st 6247   2ndc2nd 6248
This theorem is referenced by:  2ndrn  6295  1st2ndbr  6296  elopabi  6312  cnvf1olem  6344  ordpinq  8714  addassnq  8729  mulassnq  8730  distrnq  8732  mulidnq  8734  recmulnq  8735  ltexnq  8746  fsumcnv  12444  cofulid  13974  cofurid  13975  idffth  14017  cofull  14018  cofth  14019  ressffth  14022  isnat2  14032  nat1st2nd  14035  homadmcd  14084  catciso  14149  prf1st  14188  prf2nd  14189  1st2ndprf  14190  curfuncf  14222  uncfcurf  14223  curf2ndf  14231  yonffthlem  14266  yoniso  14269  dprd2dlem2  15485  dprd2dlem1  15486  dprd2da  15487  2ndcctbss  17398  caubl  18948  rngoi  21358  drngoi  21385  nvop2  21477  nvvop  21478  nvop  21556  phop  21709  cvmliftlem1  24419  heiborlem3  26043  isdrngo1  26093  iscrngo2  26129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-iota 5322  df-fun 5360  df-fv 5366  df-1st 6249  df-2nd 6250
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