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Theorem 1st2nd 6422
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 4914 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3329 . . 3  |-  ( ( B  C_  ( _V  X.  _V )  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
31, 2sylanb 460 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  ( _V  X.  _V ) )
4 1st2nd2 6415 . 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 360    = wceq 1653    e. wcel 1727   _Vcvv 2962    C_ wss 3306   <.cop 3841    X. cxp 4905   Rel wrel 4912   ` cfv 5483   1stc1st 6376   2ndc2nd 6377
This theorem is referenced by:  2ndrn  6424  1st2ndbr  6425  elopabi  6441  cnvf1olem  6473  ordpinq  8851  addassnq  8866  mulassnq  8867  distrnq  8869  mulidnq  8871  recmulnq  8872  ltexnq  8883  fsumcnv  12588  cofulid  14118  cofurid  14119  idffth  14161  cofull  14162  cofth  14163  ressffth  14166  isnat2  14176  nat1st2nd  14179  homadmcd  14228  catciso  14293  prf1st  14332  prf2nd  14333  1st2ndprf  14334  curfuncf  14366  uncfcurf  14367  curf2ndf  14375  yonffthlem  14410  yoniso  14413  dprd2dlem2  15629  dprd2dlem1  15630  dprd2da  15631  2ndcctbss  17549  utop2nei  18311  utop3cls  18312  caubl  19291  rngoi  21999  drngoi  22026  nvop2  22118  nvvop  22119  nvop  22197  phop  22350  cvmliftlem1  25003  fprodcnv  25338  heiborlem3  26560  isdrngo1  26610  iscrngo2  26646  wlkop  28359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fv 5491  df-1st 6378  df-2nd 6379
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