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Theorem 1st2nd 7972
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 5641 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
2 ssel2 3940 . . 3 ((𝐵 ⊆ (V × V) ∧ 𝐴𝐵) → 𝐴 ∈ (V × V))
31, 2sylanb 582 . 2 ((Rel 𝐵𝐴𝐵) → 𝐴 ∈ (V × V))
4 1st2nd2 7961 . 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 397   = wceq 1542  wcel 2107  Vcvv 3446  wss 3911  cop 4593   × cxp 5632  Rel wrel 5639  cfv 6497  1st c1st 7920  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  2ndrn  7974  1st2ndbr  7975  funfv1st2nd  7979  funelss  7980  elopabi  7995  cnvf1olem  8043  ordpinq  10880  addassnq  10895  mulassnq  10896  distrnq  10898  mulidnq  10900  recmulnq  10901  ltexnq  10912  fsumcnv  15659  fprodcnv  15867  cofulid  17777  cofurid  17778  idffth  17821  cofull  17822  cofth  17823  ressffth  17826  isnat2  17836  nat1st2nd  17839  homadmcd  17929  catciso  17998  prf1st  18093  prf2nd  18094  1st2ndprf  18095  curfuncf  18128  uncfcurf  18129  curf2ndf  18137  yonffthlem  18172  yoniso  18175  dprd2dlem2  19820  dprd2dlem1  19821  dprd2da  19822  mdetunilem9  21972  2ndcctbss  22809  utop2nei  23605  utop3cls  23606  caubl  24675  wlkop  28579  nvop2  29553  nvvop  29554  nvop  29621  phop  29763  fgreu  31591  1stpreimas  31622  gsumhashmul  31901  cvmliftlem1  33882  heiborlem3  36275  rngoi  36361  drngoi  36413  isdrngo1  36418  iscrngo2  36459
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