Theorem funsn 5853

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

Step	Hyp	Ref	Expression
1		funsn.1	. 2 ⊢ 𝐴 ∈ V
2		funsn.2	. 2 ⊢ 𝐵 ∈ V
3		funsng 5851	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
4	1, 2, 3	mp2an 704	1 ⊢ Fun {⟨𝐴, 𝐵⟩}

Syntax hints:   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  Fun wfun 5798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806

This theorem is referenced by:  funtp  5859  fun0  5868  funop  6320  funsndifnop  6321  fvsn  6351  wfrlem13  7314  dcomex  9152  axdc3lem4  9158  xpsc0  16043  xpsc1  16044  bnj1421  30364  funop1  40327