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Theorem funcnvsn 6618
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 6621 via cnvsn 6248, but stating it this way allows 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 6125 . 2 Rel {⟨𝐴, 𝐵⟩}
2 moeq 3716 . . . 4 ∃*𝑦 𝑦 = 𝐴
3 vex 3482 . . . . . . . 8 𝑥 ∈ V
4 vex 3482 . . . . . . . 8 𝑦 ∈ V
53, 4brcnv 5896 . . . . . . 7 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦{⟨𝐴, 𝐵⟩}𝑥)
6 df-br 5149 . . . . . . 7 (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
75, 6bitri 275 . . . . . 6 (𝑥{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8 elsni 4648 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
94, 3opth1 5486 . . . . . . 7 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
108, 9syl 17 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
117, 10sylbi 217 . . . . 5 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦 = 𝐴)
1211moimi 2543 . . . 4 (∃*𝑦 𝑦 = 𝐴 → ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦)
132, 12ax-mp 5 . . 3 ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
1413ax-gen 1792 . 2 𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
15 dffun6 6576 . 2 (Fun {⟨𝐴, 𝐵⟩} ↔ (Rel {⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦))
161, 14, 15mpbir2an 711 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:  wal 1535   = wceq 1537  wcel 2106  ∃*wmo 2536  {csn 4631  cop 4637   class class class wbr 5148  ccnv 5688  Rel wrel 5694  Fun wfun 6557
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-fun 6565
This theorem is referenced by:  funsng  6619  funcnvpr  6630  funcnvtp  6631  funcnvs1  14948  0spth  30155  funen1cnv  35081
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