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Theorem fnsng 6405
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
fnsng ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})

Proof of Theorem fnsng
StepHypRef Expression
1 funsng 6404 . 2 ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})
2 dmsnopg 6069 . . 3 (𝐵𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
32adantl 482 . 2 ((𝐴𝑉𝐵𝑊) → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4 df-fn 6357 . 2 ({⟨𝐴, 𝐵⟩} Fn {𝐴} ↔ (Fun {⟨𝐴, 𝐵⟩} ∧ dom {⟨𝐴, 𝐵⟩} = {𝐴}))
51, 3, 4sylanbrc 583 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {csn 4564  cop 4570  dom cdm 5554  Fun wfun 6348   Fn wfn 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-fun 6356  df-fn 6357
This theorem is referenced by:  fnsn  6411  fnunsn  6463  fvsnun2  6943  fsnunfv  6947  mat1dimscm  21019  m1detdiag  21141  actfunsnf1o  31780  actfunsnrndisj  31781  breprexplema  31806  noextenddif  33078  noextendlt  33079  noextendgt  33080  fnsnbt  39004
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