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Theorem fnsng 5282
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 5281 . 2 ((𝐴𝑉𝐵𝑊) → Fun {⟨𝐴, 𝐵⟩})
2 dmsnopg 5118 . . 3 (𝐵𝑊 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
32adantl 277 . 2 ((𝐴𝑉𝐵𝑊) → dom {⟨𝐴, 𝐵⟩} = {𝐴})
4 df-fn 5238 . 2 ({⟨𝐴, 𝐵⟩} Fn {𝐴} ↔ (Fun {⟨𝐴, 𝐵⟩} ∧ dom {⟨𝐴, 𝐵⟩} = {𝐴}))
51, 3, 4sylanbrc 417 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  {csn 3607  cop 3610  dom cdm 4644  Fun wfun 5229   Fn wfn 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-fun 5237  df-fn 5238
This theorem is referenced by:  fnsn  5289  fnunsn  5342  fsnunfv  5737  tfr0dm  6346  ennnfonelemhom  12465
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