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Theorem f1osn 5715
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  e. 
_V
f1osn.2  |-  B  e. 
_V
Assertion
Ref Expression
f1osn  |-  { <. A ,  B >. } : { A } -1-1-onto-> { B }

Proof of Theorem f1osn
StepHypRef Expression
1 f1osn.1 . . 3  |-  A  e. 
_V
2 f1osn.2 . . 3  |-  B  e. 
_V
31, 2fnsn 5504 . 2  |-  { <. A ,  B >. }  Fn  { A }
42, 1fnsn 5504 . . 3  |-  { <. B ,  A >. }  Fn  { B }
51, 2cnvsn 5352 . . . 4  |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
65fneq1i 5539 . . 3  |-  ( `' { <. A ,  B >. }  Fn  { B } 
<->  { <. B ,  A >. }  Fn  { B } )
74, 6mpbir 201 . 2  |-  `' { <. A ,  B >. }  Fn  { B }
8 dff1o4 5682 . 2  |-  ( {
<. A ,  B >. } : { A } -1-1-onto-> { B }  <->  ( { <. A ,  B >. }  Fn  { A }  /\  `' { <. A ,  B >. }  Fn  { B } ) )
93, 7, 8mpbir2an 887 1  |-  { <. A ,  B >. } : { A } -1-1-onto-> { B }
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2956   {csn 3814   <.cop 3817   `'ccnv 4877    Fn wfn 5449   -1-1-onto->wf1o 5453
This theorem is referenced by:  f1osng  5716  fsn  5906  mapsn  7055  ensn1  7171  phplem2  7287  isinf  7322  pssnn  7327  ac6sfi  7351  marypha1lem  7438  hashf1lem1  11704  0ram  13388  imasdsf1olem  18403  constr1trl  21588  vdegp1ai  21706  vdegp1bi  21707  grposn  21803  subfacp1lem5  24870  axlowdimlem10  25890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461
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