Theorem fnsn 6603

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 6597	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
4	1, 2, 3	mp2an 690	1 ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}

Syntax hints:   ∈ wcel 2106  Vcvv 3474  {csn 4627  ⟨cop 4633   Fn wfn 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-fun 6542  df-fn 6543

This theorem is referenced by:  f1osn  6870  fnsnb  7160  frrlem11  8277  frrlem12  8278  elixpsn  8927  axdc3lem4  10444  hashf1lem1  14411  hashf1lem1OLD  14412  axlowdimlem8  28196  axlowdimlem9  28197  axlowdimlem11  28199  axlowdimlem12  28200  bnj927  33768  cvmliftlem4  34267  cvmliftlem5  34268  finixpnum  36461  poimirlem3  36479