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Theorem f1osn 5529
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 5320 . 2  |-  { <. A ,  B >. }  Fn  { A }
42, 1fnsn 5320 . . 3  |-  { <. B ,  A >. }  Fn  { B }
51, 2cnvsn 5171 . . . 4  |-  `' { <. A ,  B >. }  =  { <. B ,  A >. }
65fneq1i 5354 . . 3  |-  ( `' { <. A ,  B >. }  Fn  { B } 
<->  { <. B ,  A >. }  Fn  { B } )
74, 6mpbir 200 . 2  |-  `' { <. A ,  B >. }  Fn  { B }
8 dff1o4 5496 . 2  |-  ( {
<. A ,  B >. } : { A } -1-1-onto-> { B }  <->  ( { <. A ,  B >. }  Fn  { A }  /\  `' { <. A ,  B >. }  Fn  { B } ) )
93, 7, 8mpbir2an 886 1  |-  { <. A ,  B >. } : { A } -1-1-onto-> { B }
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   `'ccnv 4704    Fn wfn 5266   -1-1-onto->wf1o 5270
This theorem is referenced by:  f1osng  5530  fsn  5712  mapsn  6825  ensn1  6941  phplem2  7057  isinf  7092  pssnn  7097  ac6sfi  7117  marypha1lem  7202  hashf1lem1  11409  0ram  13083  imasdsf1olem  17953  grposn  20898  subfacp1lem5  23730  vdegp1ai  23923  vdegp1bi  23924  axlowdimlem10  24651  1alg  25825  phckle  26130  psckle  26131  pgapspf  26155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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