ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fnsng GIF version

Theorem fnsng 5028
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 5027 . 2 ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})
2 dmsnopg 4870 . . 3 (𝐵𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
32adantl 271 . 2 ((𝐴𝑉𝐵𝑊) → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4 df-fn 4986 . 2 ({⟨𝐴, 𝐵⟩} Fn {𝐴} ↔ (Fun {⟨𝐴, 𝐵⟩} ∧ dom {⟨𝐴, 𝐵⟩} = {𝐴}))
51, 3, 4sylanbrc 408 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  {csn 3431  cop 3434  dom cdm 4413  Fun wfun 4977   Fn wfn 4978
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-fun 4985  df-fn 4986
This theorem is referenced by:  fnsn  5035  fnunsn  5088  fsnunfv  5462  tfr0dm  6043
  Copyright terms: Public domain W3C validator