ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1st2nd GIF version

Theorem 1st2nd 5885
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 4407 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
2 ssel2 3005 . . 3 ((𝐵 ⊆ (V × V) ∧ 𝐴𝐵) → 𝐴 ∈ (V × V))
31, 2sylanb 278 . 2 ((Rel 𝐵𝐴𝐵) → 𝐴 ∈ (V × V))
4 1st2nd2 5879 . 2 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
53, 4syl 14 1 ((Rel 𝐵𝐴𝐵) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  Vcvv 2612  wss 2984  cop 3425   × cxp 4398  Rel wrel 4405  cfv 4968  1st c1st 5843  2nd c2nd 5844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-iota 4933  df-fun 4970  df-fv 4976  df-1st 5845  df-2nd 5846
This theorem is referenced by:  2ndrn  5887  1st2ndbr  5888  elopabi  5899  cnvf1olem  5923
  Copyright terms: Public domain W3C validator