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Theorem 1st2nd 6844
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 4852 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3456 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 474 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6835 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
53, 4syl 17 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   _Vcvv 3078    C_ wss 3433   <.cop 3999    X. cxp 4843   Rel wrel 4850   ` cfv 5592   1stc1st 6796   2ndc2nd 6797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5556  df-fun 5594  df-fv 5600  df-1st 6798  df-2nd 6799
This theorem is referenced by:  2ndrn  6846  1st2ndbr  6847  elopabi  6859  cnvf1olem  6896  ordpinq  9357  addassnq  9372  mulassnq  9373  distrnq  9375  mulidnq  9377  recmulnq  9378  ltexnq  9389  fsumcnv  13801  fprodcnv  14004  cofulid  15747  cofurid  15748  idffth  15790  cofull  15791  cofth  15792  ressffth  15795  isnat2  15805  nat1st2nd  15808  homadmcd  15889  catciso  15954  prf1st  16041  prf2nd  16042  1st2ndprf  16043  curfuncf  16075  uncfcurf  16076  curf2ndf  16084  yonffthlem  16119  yoniso  16122  dprd2dlem2  17614  dprd2dlem1  17615  dprd2da  17616  mdetunilem9  19582  2ndcctbss  20407  utop2nei  21202  utop3cls  21203  caubl  22196  elusuhgra  24982  wlkop  25142  rngoi  25994  drngoi  26021  nvop2  26113  nvvop  26114  nvop  26192  phop  26345  fgreu  28155  1stpreimas  28167  cvmliftlem1  29837  heiborlem3  31893  isdrngo1  31943  iscrngo2  31979  fusgraimpcl  38554  fusgraimpclALT  38556  fusgraimpclALT2  38558  usgfis  38573  usgfisALT  38577
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