Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmtrfb Unicode version

Theorem pmtrfb 27406
Description: An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t  |-  T  =  (pmTrsp `  D )
pmtrrn.r  |-  R  =  ran  T
Assertion
Ref Expression
pmtrfb  |-  ( F  e.  R  <->  ( D  e.  _V  /\  F : D
-1-1-onto-> D  /\  dom  ( F 
\  _I  )  ~~  2o ) )

Proof of Theorem pmtrfb
StepHypRef Expression
1 pmtrrn.t . . . . 5  |-  T  =  (pmTrsp `  D )
2 pmtrrn.r . . . . 5  |-  R  =  ran  T
3 eqid 2283 . . . . 5  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 27400 . . . 4  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
5 simpl1 958 . . . 4  |-  ( ( ( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) )  ->  D  e.  _V )
64, 5syl 15 . . 3  |-  ( F  e.  R  ->  D  e.  _V )
71, 2pmtrff1o 27404 . . 3  |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
8 simpl3 960 . . . 4  |-  ( ( ( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) )  ->  dom  ( F  \  _I  )  ~~  2o )
94, 8syl 15 . . 3  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
106, 7, 93jca 1132 . 2  |-  ( F  e.  R  ->  ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o ) )
11 simp2 956 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F : D
-1-1-onto-> D )
12 difss 3303 . . . . . . . 8  |-  ( F 
\  _I  )  C_  F
13 dmss 4878 . . . . . . . 8  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
1412, 13ax-mp 8 . . . . . . 7  |-  dom  ( F  \  _I  )  C_  dom  F
15 f1odm 5476 . . . . . . 7  |-  ( F : D -1-1-onto-> D  ->  dom  F  =  D )
1614, 15syl5sseq 3226 . . . . . 6  |-  ( F : D -1-1-onto-> D  ->  dom  ( F 
\  _I  )  C_  D )
171, 2pmtrrn 27399 . . . . . 6  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `  dom  ( F 
\  _I  ) )  e.  R )
1816, 17syl3an2 1216 . . . . 5  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `
 dom  ( F  \  _I  ) )  e.  R )
191, 2pmtrff1o 27404 . . . . 5  |-  ( ( T `  dom  ( F  \  _I  ) )  e.  R  ->  ( T `  dom  ( F 
\  _I  ) ) : D -1-1-onto-> D )
2018, 19syl 15 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `
 dom  ( F  \  _I  ) ) : D -1-1-onto-> D )
21 simp3 957 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  ( F  \  _I  )  ~~  2o )
221pmtrmvd 27398 . . . . 5  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  ( ( T `  dom  ( F  \  _I  ) )  \  _I  )  =  dom  ( F 
\  _I  ) )
2316, 22syl3an2 1216 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  (
( T `  dom  ( F  \  _I  )
)  \  _I  )  =  dom  ( F  \  _I  ) )
24 f1otrspeq 27390 . . . 4  |-  ( ( ( F : D -1-1-onto-> D  /\  ( T `  dom  ( F  \  _I  )
) : D -1-1-onto-> D )  /\  ( dom  ( F  \  _I  )  ~~  2o  /\  dom  ( ( T `  dom  ( F  \  _I  ) ) 
\  _I  )  =  dom  ( F  \  _I  ) ) )  ->  F  =  ( T `  dom  ( F  \  _I  ) ) )
2511, 20, 21, 23, 24syl22anc 1183 . . 3  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F  =  ( T `  dom  ( F  \  _I  )
) )
2625, 18eqeltrd 2357 . 2  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F  e.  R )
2710, 26impbii 180 1  |-  ( F  e.  R  <->  ( D  e.  _V  /\  F : D
-1-1-onto-> D  /\  dom  ( F 
\  _I  )  ~~  2o ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   class class class wbr 4023    _I cid 4304   dom cdm 4689   ran crn 4690   -1-1-onto->wf1o 5254   ` cfv 5255   2oc2o 6473    ~~ cen 6860  pmTrspcpmtr 27384
This theorem is referenced by:  pmtrfconj  27407  symggen  27411  psgnunilem1  27416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pmtr 27385
  Copyright terms: Public domain W3C validator