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Theorem fnsn 6607
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 6601 . 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 3475  {csn 4629  cop 4635   Fn wfn 6539
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 5300  ax-nul 5307  ax-pr 5428
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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-fun 6546  df-fn 6547
This theorem is referenced by:  f1osn  6874  fnsnb  7164  frrlem11  8281  frrlem12  8282  elixpsn  8931  axdc3lem4  10448  hashf1lem1  14415  hashf1lem1OLD  14416  axlowdimlem8  28207  axlowdimlem9  28208  axlowdimlem11  28210  axlowdimlem12  28211  bnj927  33780  cvmliftlem4  34279  cvmliftlem5  34280  finixpnum  36473  poimirlem3  36491
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