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Theorem funsn 6601
Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
Hypotheses
Ref Expression
funsn.1 𝐴 ∈ V
funsn.2 𝐵 ∈ V
Assertion
Ref Expression
funsn Fun {⟨𝐴, 𝐵⟩}

Proof of Theorem funsn
StepHypRef Expression
1 funsn.1 . 2 𝐴 ∈ V
2 funsn.2 . 2 𝐵 ∈ V
3 funsng 6599 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
41, 2, 3mp2an 690 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3474  {csn 4628  cop 4634  Fun wfun 6537
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545
This theorem is referenced by:  funtp  6605  fun0  6613  funop  7146  funsndifnop  7148  wfrlem13OLD  8320  dcomex  10441  axdc3lem4  10447  cnfldfunALT  20956  cnfldfunALTOLD  20957  bnj1421  34048  funop1  45981
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