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

Theorem funcnvsn 5012
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5015 via cnvsn 4867, but stating it this way allows us to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.)
Assertion
Ref Expression
funcnvsn Fun {⟨𝐴, 𝐵⟩}

Proof of Theorem funcnvsn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4765 . 2 Rel {⟨𝐴, 𝐵⟩}
2 moeq 2778 . . . 4 ∃*𝑦 𝑦 = 𝐴
3 vex 2615 . . . . . . . 8 𝑥 ∈ V
4 vex 2615 . . . . . . . 8 𝑦 ∈ V
53, 4brcnv 4577 . . . . . . 7 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦{⟨𝐴, 𝐵⟩}𝑥)
6 df-br 3812 . . . . . . 7 (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
75, 6bitri 182 . . . . . 6 (𝑥{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8 elsni 3440 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
94, 3opth1 4027 . . . . . . 7 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
108, 9syl 14 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
117, 10sylbi 119 . . . . 5 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦 = 𝐴)
1211moimi 2008 . . . 4 (∃*𝑦 𝑦 = 𝐴 → ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦)
132, 12ax-mp 7 . . 3 ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
1413ax-gen 1379 . 2 𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
15 dffun6 4983 . 2 (Fun {⟨𝐴, 𝐵⟩} ↔ (Rel {⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦))
161, 14, 15mpbir2an 884 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff set class
Syntax hints:  wal 1283   = wceq 1285  wcel 1434  ∃*wmo 1944  {csn 3422  cop 3425   class class class wbr 3811  ccnv 4400  Rel wrel 4406  Fun wfun 4963
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-fun 4971
This theorem is referenced by:  funsng  5013
  Copyright terms: Public domain W3C validator