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Theorem 1st2nd 6733
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 4958 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
2 ssel2 3462 . . 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 6726 . 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 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3439   <.cop 3994    X. cxp 4949   Rel wrel 4956   ` cfv 5529   1stc1st 6688   2ndc2nd 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-iota 5492  df-fun 5531  df-fv 5537  df-1st 6690  df-2nd 6691
This theorem is referenced by:  2ndrn  6735  1st2ndbr  6736  elopabi  6748  cnvf1olem  6783  ordpinq  9226  addassnq  9241  mulassnq  9242  distrnq  9244  mulidnq  9246  recmulnq  9247  ltexnq  9258  fsumcnv  13361  cofulid  14922  cofurid  14923  idffth  14965  cofull  14966  cofth  14967  ressffth  14970  isnat2  14980  nat1st2nd  14983  homadmcd  15032  catciso  15097  prf1st  15136  prf2nd  15137  1st2ndprf  15138  curfuncf  15170  uncfcurf  15171  curf2ndf  15179  yonffthlem  15214  yoniso  15217  dprd2dlem2  16664  dprd2dlem1  16665  dprd2da  16666  mdetunilem9  18561  2ndcctbss  19194  utop2nei  19960  utop3cls  19961  caubl  20953  rngoi  24039  drngoi  24066  nvop2  24158  nvvop  24159  nvop  24237  phop  24390  fgreu  26161  cvmliftlem1  27338  fprodcnv  27658  heiborlem3  28880  isdrngo1  28930  iscrngo2  28966  wlkop  30450
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