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Theorem 1st2nd 7159
 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 5081 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
2 ssel2 3578 . . 3 ((𝐵 ⊆ (V × V) ∧ 𝐴𝐵) → 𝐴 ∈ (V × V))
31, 2sylanb 489 . 2 ((Rel 𝐵𝐴𝐵) → 𝐴 ∈ (V × V))
4 1st2nd2 7150 . 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 384   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ⊆ wss 3555  ⟨cop 4154   × cxp 5072  Rel wrel 5079  ‘cfv 5847  1st c1st 7111  2nd c2nd 7112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fv 5855  df-1st 7113  df-2nd 7114 This theorem is referenced by:  2ndrn  7161  1st2ndbr  7162  elopabi  7176  cnvf1olem  7220  ordpinq  9709  addassnq  9724  mulassnq  9725  distrnq  9727  mulidnq  9729  recmulnq  9730  ltexnq  9741  fsumcnv  14432  fprodcnv  14638  cofulid  16471  cofurid  16472  idffth  16514  cofull  16515  cofth  16516  ressffth  16519  isnat2  16529  nat1st2nd  16532  homadmcd  16613  catciso  16678  prf1st  16765  prf2nd  16766  1st2ndprf  16767  curfuncf  16799  uncfcurf  16800  curf2ndf  16808  yonffthlem  16843  yoniso  16846  dprd2dlem2  18360  dprd2dlem1  18361  dprd2da  18362  mdetunilem9  20345  2ndcctbss  21168  utop2nei  21964  utop3cls  21965  caubl  23014  wlkop  26393  nvop2  27309  nvvop  27310  nvop  27377  phop  27519  fgreu  29311  1stpreimas  29323  cvmliftlem1  30972  heiborlem3  33241  rngoi  33327  drngoi  33379  isdrngo1  33384  iscrngo2  33425
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