MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcnvsn Structured version   Visualization version   GIF version

Theorem funcnvsn 6598
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 6601 via cnvsn 6225, 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 6103 . 2 Rel {⟨𝐴, 𝐵⟩}
2 moeq 3703 . . . 4 ∃*𝑦 𝑦 = 𝐴
3 vex 3478 . . . . . . . 8 𝑥 ∈ V
4 vex 3478 . . . . . . . 8 𝑦 ∈ V
53, 4brcnv 5882 . . . . . . 7 (𝑥{⟨𝐴, 𝐵⟩}𝑦𝑦{⟨𝐴, 𝐵⟩}𝑥)
6 df-br 5149 . . . . . . 7 (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
75, 6bitri 274 . . . . . 6 (𝑥{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8 elsni 4645 . . . . . . 7 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
94, 3opth1 5475 . . . . . . 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 2539 . . . 4 (∃*𝑦 𝑦 = 𝐴 → ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦)
132, 12ax-mp 5 . . 3 ∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
1413ax-gen 1797 . 2 𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦
15 dffun6 6556 . 2 (Fun {⟨𝐴, 𝐵⟩} ↔ (Rel {⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥{⟨𝐴, 𝐵⟩}𝑦))
161, 14, 15mpbir2an 709 1 Fun {⟨𝐴, 𝐵⟩}
Colors of variables: wff setvar class
Syntax hints:  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2532  {csn 4628  cop 4634   class class class wbr 5148  ccnv 5675  Rel wrel 5681  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545
This theorem is referenced by:  funsng  6599  funcnvpr  6610  funcnvtp  6611  funcnvs1  14862  0spth  29376  funen1cnv  34086
  Copyright terms: Public domain W3C validator