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Theorem funsn 6621
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 6619 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
41, 2, 3mp2an 692 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  {csn 4631  cop 4637  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565
This theorem is referenced by:  funtp  6625  fun0  6633  funop  7169  funsndifnop  7171  wfrlem13OLD  8360  dcomex  10485  axdc3lem4  10491  cnfldfunALT  21397  cnfldfunALTOLD  21410  cnfldfunALTOLDOLD  21411  bnj1421  35035  funop1  47233
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