Theorem fnsn 6540

Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
fnsn.1	⊢ 𝐴 ∈ V
fnsn.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
fnsn	⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}

Proof of Theorem fnsn

Step	Hyp	Ref	Expression
1		fnsn.1	. 2 ⊢ 𝐴 ∈ V
2		fnsn.2	. 2 ⊢ 𝐵 ∈ V
3		fnsng 6534	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
4	1, 2, 3	mp2an 692	1 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}

Syntax hints:   ∈ wcel 2109  Vcvv 3436  {csn 4577  ⟨cop 4583   Fn wfn 6477

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-fun 6484  df-fn 6485

This theorem is referenced by:  f1osn  6804  fnsnbOLD  7102  frrlem11  8229  frrlem12  8230  elixpsn  8864  axdc3lem4  10347  hashf1lem1  14362  axlowdimlem8  28898  axlowdimlem9  28899  axlowdimlem11  28901  axlowdimlem12  28902  bnj927  34752  cvmliftlem4  35281  cvmliftlem5  35282  finixpnum  37605  poimirlem3  37623