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Theorem mptrel 4637
Description: The maps-to notation always describes a relationship. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
mptrel Rel (𝑥𝐴𝐵)

Proof of Theorem mptrel
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 3961 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
21relopabi 4635 1 Rel (𝑥𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wcel 1465  cmpt 3959  Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-mpt 3961  df-xp 4515  df-rel 4516
This theorem is referenced by:  cnprcl2k  12302  psmetrel  12418  metrel  12438  xmetrel  12439  xmetf  12446  mopnrel  12537
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