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Theorem funsng 6600
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
Assertion
Ref Expression
funsng ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})

Proof of Theorem funsng
StepHypRef Expression
1 funcnvsn 6599 . 2 Fun {⟨𝐵, 𝐴⟩}
2 cnvsng 6223 . . . 4 ((𝐵𝑊𝐴𝑉) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
32ancoms 460 . . 3 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
43funeqd 6571 . 2 ((𝐴𝑉𝐵𝑊) → (Fun {⟨𝐵, 𝐴⟩} ↔ Fun {⟨𝐴, 𝐵⟩}))
51, 4mpbii 232 1 ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {csn 4629  cop 4635  ccnv 5676  Fun wfun 6538
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-fun 6546
This theorem is referenced by:  fnsng  6601  funsn  6602  funprg  6603  funtpg  6604  fvsng  7178  tfrlem10  8387  snopfsupp  9386  funsnfsupp  9387  strle1  17091  setsfun  17104  setsfun0  17105  noextend  27169  p1evtxdeqlem  28769  trlsegvdeglem3  29475  bnj519  33747  bnj150  33887
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