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Theorem fnsn 6158
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 6152 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
41, 2, 3mp2an 684 1 {⟨𝐴, 𝐵⟩} Fn {𝐴}
Colors of variables: wff setvar class
Syntax hints:  wcel 2157  Vcvv 3385  {csn 4368  cop 4374   Fn wfn 6096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-fun 6103  df-fn 6104
This theorem is referenced by:  f1osn  6395  fnsnb  6661  fvsnun2  6678  elixpsn  8187  axdc3lem4  9563  hashf1lem1  13488  axlowdimlem8  26186  axlowdimlem9  26187  axlowdimlem11  26189  axlowdimlem12  26190  bnj927  31356  cvmliftlem4  31787  cvmliftlem5  31788  finixpnum  33883  poimirlem3  33901
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