MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnsng Structured version   Visualization version   GIF version

Theorem fnsng 6620
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 6619 . 2 ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})
2 dmsnopg 6235 . . 3 (𝐵𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
32adantl 481 . 2 ((𝐴𝑉𝐵𝑊) → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4 df-fn 6566 . 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 395   = wceq 1537  wcel 2106  {csn 4631  cop 4637  dom cdm 5689  Fun wfun 6557   Fn wfn 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-fun 6565  df-fn 6566
This theorem is referenced by:  fnsn  6626  fnunop  6685  fvsnun2  7203  fsnunfv  7207  mat1dimscm  22497  m1detdiag  22619  noextenddif  27728  noextendlt  27729  noextendgt  27730  actfunsnf1o  34598  actfunsnrndisj  34599  breprexplema  34624  sticksstones11  42138  metakunt19  42205  fnsnbt  42250
  Copyright terms: Public domain W3C validator