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Theorem funcnvsn 6547
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 6550 via cnvsn 6175, 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 6053 . 2 Rel {⟨𝐴, 𝐵⟩}
2 moeq 3664 . . . 4 ∃*𝑦 𝑦 = 𝐴
3 vex 3448 . . . . . . . 8 𝑥 ∈ V
4 vex 3448 . . . . . . . 8 𝑦 ∈ V
53, 4brcnv 5835 . . . . . . 7 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦{⟨𝐴, 𝐵⟩}𝑥)
6 df-br 5105 . . . . . . 7 (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
75, 6bitri 275 . . . . . 6 (𝑥{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8 elsni 4602 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
94, 3opth1 5431 . . . . . . 7 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
108, 9syl 17 . . . . . 6 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
117, 10sylbi 216 . . . . 5 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦 = 𝐴)
1211moimi 2545 . . . 4 (∃*𝑦 𝑦 = 𝐴 → ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦)
132, 12ax-mp 5 . . 3 ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
1413ax-gen 1798 . 2 𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
15 dffun6 6505 . 2 (Fun {⟨𝐴, 𝐵⟩} ↔ (Rel {⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦))
161, 14, 15mpbir2an 710 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:  wal 1540   = wceq 1542  wcel 2107  ∃*wmo 2538  {csn 4585  cop 4591   class class class wbr 5104  ccnv 5630  Rel wrel 5636  Fun wfun 6486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-fun 6494
This theorem is referenced by:  funsng  6548  funcnvpr  6559  funcnvtp  6560  funcnvs1  14734  0spth  28875  funen1cnv  33472
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