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Theorem f1osn 6889
Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
f1osn.1 𝐴 ∈ V
f1osn.2 𝐵 ∈ V
Assertion
Ref Expression
f1osn {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}

Proof of Theorem f1osn
StepHypRef Expression
1 f1osn.1 . . 3 𝐴 ∈ V
2 f1osn.2 . . 3 𝐵 ∈ V
31, 2fnsn 6626 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐴}
42, 1fnsn 6626 . . 3 {⟨𝐵, 𝐴⟩} Fn {𝐵}
51, 2cnvsn 6248 . . . 4 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
65fneq1i 6666 . . 3 ({⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
74, 6mpbir 231 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐵}
8 dff1o4 6857 . 2 ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ {⟨𝐴, 𝐵⟩} Fn {𝐵}))
93, 7, 8mpbir2an 711 1 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  {csn 4631  cop 4637  ccnv 5688   Fn wfn 6558  1-1-ontowf1o 6562
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-dm 5699  df-rn 5700  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  f1osng  6890  fsn  7155  ensn1  9060  pssnn  9207  phplem2OLD  9253  isinf  9294  isinfOLD  9295  ac6sfi  9318  marypha1lem  9471  hashf1lem1  14491  0ram  17054  mdet0f1o  22615  imasdsf1olem  24399  istrkg2ld  28483  axlowdimlem10  28981  subfacp1lem5  35169  poimirlem3  37610  grposnOLD  37869
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