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Theorem fnsn 6560
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
fnsn.1 𝐴 ∈ V
fnsn.2 𝐵 ∈ V
Assertion
Ref Expression
fnsn {⟨𝐴, 𝐵⟩} Fn {𝐴}

Proof of Theorem fnsn
StepHypRef Expression
1 fnsn.1 . 2 𝐴 ∈ V
2 fnsn.2 . 2 𝐵 ∈ V
3 fnsng 6554 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
41, 2, 3mp2an 691 1 {⟨𝐴, 𝐵⟩} Fn {𝐴}
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3444  {csn 4587  cop 4593   Fn wfn 6492
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-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  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-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-fun 6499  df-fn 6500
This theorem is referenced by:  f1osn  6825  fnsnb  7113  frrlem11  8228  frrlem12  8229  elixpsn  8878  axdc3lem4  10394  hashf1lem1  14359  hashf1lem1OLD  14360  axlowdimlem8  27940  axlowdimlem9  27941  axlowdimlem11  27943  axlowdimlem12  27944  bnj927  33438  cvmliftlem4  33939  cvmliftlem5  33940  finixpnum  36109  poimirlem3  36127
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