Theorem fnsn 6577

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 6571	. 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 3450  {csn 4592  ⟨cop 4598   Fn wfn 6509

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-fun 6516  df-fn 6517

This theorem is referenced by:  f1osn  6843  fnsnbOLD  7143  frrlem11  8278  frrlem12  8279  elixpsn  8913  axdc3lem4  10413  hashf1lem1  14427  axlowdimlem8  28883  axlowdimlem9  28884  axlowdimlem11  28886  axlowdimlem12  28887  bnj927  34766  cvmliftlem4  35282  cvmliftlem5  35283  finixpnum  37606  poimirlem3  37624