Theorem funsn 6619

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 6617	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → Fun {⟨𝐴, 𝐵⟩})
4	1, 2, 3	mp2an 692	1 ⊢ Fun {⟨𝐴, 𝐵⟩}

Syntax hints:   ∈ wcel 2108  Vcvv 3480  {csn 4626  ⟨cop 4632  Fun wfun 6555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-fun 6563

This theorem is referenced by:  funtp  6623  fun0  6631  funop  7169  funsndifnop  7171  wfrlem13OLD  8361  dcomex  10487  axdc3lem4  10493  cnfldfunALT  21379  cnfldfunALTOLD  21392  cnfldfunALTOLDOLD  21393  bnj1421  35056  funop1  47295