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Theorem 1st2nd 8063
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 𝐵𝐴𝐵) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)

Proof of Theorem 1st2nd
StepHypRef Expression
1 df-rel 5696 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
2 ssel2 3990 . . 3 ((𝐵 ⊆ (V × V) ∧ 𝐴𝐵) → 𝐴 ∈ (V × V))
31, 2sylanb 581 . 2 ((Rel 𝐵𝐴𝐵) → 𝐴 ∈ (V × V))
4 1st2nd2 8052 . 2 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
53, 4syl 17 1 ((Rel 𝐵𝐴𝐵) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  cop 4637   × cxp 5687  Rel wrel 5694  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  2ndrn  8065  1st2ndbr  8066  funfv1st2nd  8070  funelss  8071  elopabi  8086  cnvf1olem  8134  ordpinq  10981  addassnq  10996  mulassnq  10997  distrnq  10999  mulidnq  11001  recmulnq  11002  ltexnq  11013  fsumcnv  15806  fprodcnv  16016  cofulid  17941  cofurid  17942  idffth  17987  cofull  17988  cofth  17989  ressffth  17992  isnat2  18003  nat1st2nd  18006  homadmcd  18096  catciso  18165  prf1st  18260  prf2nd  18261  1st2ndprf  18262  curfuncf  18295  uncfcurf  18296  curf2ndf  18304  yonffthlem  18339  yoniso  18342  dprd2dlem2  20075  dprd2dlem1  20076  dprd2da  20077  mdetunilem9  22642  2ndcctbss  23479  utop2nei  24275  utop3cls  24276  caubl  25356  wlkop  29661  nvop2  30637  nvvop  30638  nvop  30705  phop  30847  fgreu  32689  1stpreimas  32721  gsumhashmul  33047  cvmliftlem1  35270  heiborlem3  37800  rngoi  37886  drngoi  37938  isdrngo1  37943  iscrngo2  37984
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