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Theorem f1osn 6214
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 5984 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐴}
42, 1fnsn 5984 . . 3 {⟨𝐵, 𝐴⟩} Fn {𝐵}
51, 2cnvsn 5655 . . . 4 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
65fneq1i 6023 . . 3 ({⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
74, 6mpbir 221 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐵}
8 dff1o4 6183 . 2 ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ {⟨𝐴, 𝐵⟩} Fn {𝐵}))
93, 7, 8mpbir2an 975 1 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231  {csn 4210  cop 4216  ccnv 5142   Fn wfn 5921  1-1-ontowf1o 5925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933
This theorem is referenced by:  f1osng  6215  fsn  6442  mapsn  7941  ensn1  8061  phplem2  8181  isinf  8214  pssnn  8219  ac6sfi  8245  marypha1lem  8380  hashf1lem1  13277  0ram  15771  mdet0f1o  20447  imasdsf1olem  22225  istrkg2ld  25404  axlowdimlem10  25876  subfacp1lem5  31292  poimirlem3  33542  grposnOLD  33811
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