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Theorem fnsn 5934
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 5926 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
41, 2, 3mp2an 707 1 {⟨𝐴, 𝐵⟩} Fn {𝐴}
Colors of variables: wff setvar class
Syntax hints:  wcel 1988  Vcvv 3195  {csn 4168  cop 4174   Fn wfn 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-fun 5878  df-fn 5879
This theorem is referenced by:  f1osn  6163  fnsnb  6417  fvsnun2  6434  elixpsn  7932  axdc3lem4  9260  hashf1lem1  13222  axlowdimlem8  25810  axlowdimlem9  25811  axlowdimlem11  25813  axlowdimlem12  25814  bnj927  30813  cvmliftlem4  31244  cvmliftlem5  31245  finixpnum  33365  poimirlem3  33383
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