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Theorem 1st2nd 6831
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 5006 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3499 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 472 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6822 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    C_ wss 3476   <.cop 4033    X. cxp 4997   Rel wrel 5004   ` cfv 5588   1stc1st 6783   2ndc2nd 6784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fv 5596  df-1st 6785  df-2nd 6786
This theorem is referenced by:  2ndrn  6833  1st2ndbr  6834  elopabi  6846  cnvf1olem  6882  ordpinq  9322  addassnq  9337  mulassnq  9338  distrnq  9340  mulidnq  9342  recmulnq  9343  ltexnq  9354  fsumcnv  13554  cofulid  15120  cofurid  15121  idffth  15163  cofull  15164  cofth  15165  ressffth  15168  isnat2  15178  nat1st2nd  15181  homadmcd  15230  catciso  15295  prf1st  15334  prf2nd  15335  1st2ndprf  15336  curfuncf  15368  uncfcurf  15369  curf2ndf  15377  yonffthlem  15412  yoniso  15415  dprd2dlem2  16903  dprd2dlem1  16904  dprd2da  16905  mdetunilem9  18929  2ndcctbss  19762  utop2nei  20580  utop3cls  20581  caubl  21573  elusuhgra  24141  wlkop  24301  rngoi  25155  drngoi  25182  nvop2  25274  nvvop  25275  nvop  25353  phop  25506  fgreu  27282  cvmliftlem1  28481  fprodcnv  28966  heiborlem3  30139  isdrngo1  30189  iscrngo2  30225  fusgraimpcl  32121  fusgraimpclALT  32123  fusgraimpclALT2  32125  usgfis  32140  usgfisALT  32144
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