MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1osn Structured version   Visualization version   GIF version

Theorem f1osn 6821
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 6556 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐴}
42, 1fnsn 6556 . . 3 {⟨𝐵, 𝐴⟩} Fn {𝐵}
51, 2cnvsn 6176 . . . 4 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
65fneq1i 6596 . . 3 ({⟨𝐴, 𝐵⟩} Fn {𝐵} ↔ {⟨𝐵, 𝐴⟩} Fn {𝐵})
74, 6mpbir 230 . 2 {⟨𝐴, 𝐵⟩} Fn {𝐵}
8 dff1o4 6789 . 2 ({⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵} ↔ ({⟨𝐴, 𝐵⟩} Fn {𝐴} ∧ {⟨𝐴, 𝐵⟩} Fn {𝐵}))
93, 7, 8mpbir2an 709 1 {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3443  {csn 4584  cop 4590  ccnv 5630   Fn wfn 6488  1-1-ontowf1o 6492
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 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
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 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500
This theorem is referenced by:  f1osng  6822  fsn  7077  ensn1  8919  ensn1OLD  8920  pssnn  9070  phplem2OLD  9120  isinf  9162  isinfOLD  9163  pssnnOLD  9167  ac6sfi  9189  marypha1lem  9327  hashf1lem1  14306  hashf1lem1OLD  14307  0ram  16851  mdet0f1o  21893  imasdsf1olem  23677  istrkg2ld  27230  axlowdimlem10  27728  subfacp1lem5  33581  poimirlem3  36012  grposnOLD  36272
  Copyright terms: Public domain W3C validator