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Theorem funsn 5907
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 5905 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
41, 2, 3mp2an 707 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  Vcvv 3190  {csn 4155  cop 4161  Fun wfun 5851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-fun 5859
This theorem is referenced by:  funtp  5913  fun0  5922  funop  6379  funsndifnop  6381  fvsn  6411  wfrlem13  7387  dcomex  9229  axdc3lem4  9235  xpsc0  16160  xpsc1  16161  cnfldfun  19698  bnj1421  30871  funop1  40629
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