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Theorem funsng 6617
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 6616 . 2 Fun {⟨𝐵, 𝐴⟩}
2 cnvsng 6243 . . . 4 ((𝐵𝑊𝐴𝑉) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
32ancoms 458 . . 3 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
43funeqd 6588 . 2 ((𝐴𝑉𝐵𝑊) → (Fun {⟨𝐵, 𝐴⟩} ↔ Fun {⟨𝐴, 𝐵⟩}))
51, 4mpbii 233 1 ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {csn 4626  cop 4632  ccnv 5684  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-fun 6563
This theorem is referenced by:  fnsng  6618  funsn  6619  funprg  6620  funtpg  6621  fvsng  7200  tfrlem10  8427  snopfsupp  9431  funsnfsupp  9432  strle1  17195  setsfun  17208  setsfun0  17209  noextend  27711  p1evtxdeqlem  29530  trlsegvdeglem3  30241  bnj519  34750  bnj150  34890
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