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Theorem funsng 6379
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 6378 . 2 Fun {⟨𝐵, 𝐴⟩}
2 cnvsng 6051 . . . 4 ((𝐵𝑊𝐴𝑉) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
32ancoms 462 . . 3 ((𝐴𝑉𝐵𝑊) → {⟨𝐵, 𝐴⟩} = {⟨𝐴, 𝐵⟩})
43funeqd 6350 . 2 ((𝐴𝑉𝐵𝑊) → (Fun {⟨𝐵, 𝐴⟩} ↔ Fun {⟨𝐴, 𝐵⟩}))
51, 4mpbii 236 1 ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  {csn 4528  cop 4534  ccnv 5522  Fun wfun 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-fun 6330
This theorem is referenced by:  fnsng  6380  funsn  6381  funprg  6382  funtpg  6383  fvsng  6923  tfrlem10  8010  snopfsupp  8844  funsnfsupp  8845  setsfun  16513  setsfun0  16514  strle1  16587  p1evtxdeqlem  27305  trlsegvdeglem3  28010  bnj519  32114  bnj150  32256  noextend  33281
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