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Theorem 1st2nd 7362
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 5256 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
2 ssel2 3745 . . 3 ((𝐵 ⊆ (V × V) ∧ 𝐴𝐵) → 𝐴 ∈ (V × V))
31, 2sylanb 562 . 2 ((Rel 𝐵𝐴𝐵) → 𝐴 ∈ (V × V))
4 1st2nd2 7353 . 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 382   = wceq 1630  wcel 2144  Vcvv 3349  wss 3721  cop 4320   × cxp 5247  Rel wrel 5254  cfv 6031  1st c1st 7312  2nd c2nd 7313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7314  df-2nd 7315
This theorem is referenced by:  2ndrn  7364  1st2ndbr  7365  elopabi  7380  cnvf1olem  7425  ordpinq  9966  addassnq  9981  mulassnq  9982  distrnq  9984  mulidnq  9986  recmulnq  9987  ltexnq  9998  fsumcnv  14711  fprodcnv  14919  cofulid  16756  cofurid  16757  idffth  16799  cofull  16800  cofth  16801  ressffth  16804  isnat2  16814  nat1st2nd  16817  homadmcd  16898  catciso  16963  prf1st  17051  prf2nd  17052  1st2ndprf  17053  curfuncf  17085  uncfcurf  17086  curf2ndf  17094  yonffthlem  17129  yoniso  17132  dprd2dlem2  18646  dprd2dlem1  18647  dprd2da  18648  mdetunilem9  20643  2ndcctbss  21478  utop2nei  22273  utop3cls  22274  caubl  23324  wlkop  26757  nvop2  27797  nvvop  27798  nvop  27865  phop  28007  fgreu  29805  1stpreimas  29817  cvmliftlem1  31599  heiborlem3  33937  rngoi  34023  drngoi  34075  isdrngo1  34080  iscrngo2  34121
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