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Theorem fnsn 6099
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 6091 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
41, 2, 3mp2an 710 1 {⟨𝐴, 𝐵⟩} Fn {𝐴}
Colors of variables: wff setvar class
Syntax hints:  wcel 2131  Vcvv 3332  {csn 4313  cop 4319   Fn wfn 6036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-fun 6043  df-fn 6044
This theorem is referenced by:  f1osn  6329  fnsnb  6588  fvsnun2  6605  elixpsn  8105  axdc3lem4  9459  hashf1lem1  13423  axlowdimlem8  26020  axlowdimlem9  26021  axlowdimlem11  26023  axlowdimlem12  26024  bnj927  31138  cvmliftlem4  31569  cvmliftlem5  31570  finixpnum  33699  poimirlem3  33717
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