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Theorem funcnvsn 5168
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5171 via cnvsn 5021, 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 4917 . 2 Rel {⟨𝐴, 𝐵⟩}
2 moeq 2859 . . . 4 ∃*𝑦 𝑦 = 𝐴
3 vex 2689 . . . . . . . 8 𝑥 ∈ V
4 vex 2689 . . . . . . . 8 𝑦 ∈ V
53, 4brcnv 4722 . . . . . . 7 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦{⟨𝐴, 𝐵⟩}𝑥)
6 df-br 3930 . . . . . . 7 (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
75, 6bitri 183 . . . . . 6 (𝑥{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8 elsni 3545 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
94, 3opth1 4158 . . . . . . 7 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
108, 9syl 14 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
117, 10sylbi 120 . . . . 5 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦 = 𝐴)
1211moimi 2064 . . . 4 (∃*𝑦 𝑦 = 𝐴 → ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦)
132, 12ax-mp 5 . . 3 ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
1413ax-gen 1425 . 2 𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
15 dffun6 5137 . 2 (Fun {⟨𝐴, 𝐵⟩} ↔ (Rel {⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦))
161, 14, 15mpbir2an 926 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff set class
Syntax hints:  wal 1329   = wceq 1331  wcel 1480  ∃*wmo 2000  {csn 3527  cop 3530   class class class wbr 3929  ccnv 4538  Rel wrel 4544  Fun wfun 5117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-fun 5125
This theorem is referenced by:  funsng  5169
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