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Theorem pmtrfb 27282
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 2412 . . . . 5  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 27276 . . . 4  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
5 simpl1 960 . . . 4  |-  ( ( ( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) )  ->  D  e.  _V )
64, 5syl 16 . . 3  |-  ( F  e.  R  ->  D  e.  _V )
71, 2pmtrff1o 27280 . . 3  |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
8 simpl3 962 . . . 4  |-  ( ( ( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) )  ->  dom  ( F  \  _I  )  ~~  2o )
94, 8syl 16 . . 3  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
106, 7, 93jca 1134 . 2  |-  ( F  e.  R  ->  ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o ) )
11 simp2 958 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F : D
-1-1-onto-> D )
12 difss 3442 . . . . . . . 8  |-  ( F 
\  _I  )  C_  F
13 dmss 5036 . . . . . . . 8  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
1412, 13ax-mp 8 . . . . . . 7  |-  dom  ( F  \  _I  )  C_  dom  F
15 f1odm 5645 . . . . . . 7  |-  ( F : D -1-1-onto-> D  ->  dom  F  =  D )
1614, 15syl5sseq 3364 . . . . . 6  |-  ( F : D -1-1-onto-> D  ->  dom  ( F 
\  _I  )  C_  D )
171, 2pmtrrn 27275 . . . . . 6  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `  dom  ( F 
\  _I  ) )  e.  R )
1816, 17syl3an2 1218 . . . . 5  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `
 dom  ( F  \  _I  ) )  e.  R )
191, 2pmtrff1o 27280 . . . . 5  |-  ( ( T `  dom  ( F  \  _I  ) )  e.  R  ->  ( T `  dom  ( F 
\  _I  ) ) : D -1-1-onto-> D )
2018, 19syl 16 . . . 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 959 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  ( F  \  _I  )  ~~  2o )
221pmtrmvd 27274 . . . . 5  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  ( ( T `  dom  ( F  \  _I  ) )  \  _I  )  =  dom  ( F 
\  _I  ) )
2316, 22syl3an2 1218 . . . 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 27266 . . . 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 1185 . . 3  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F  =  ( T `  dom  ( F  \  _I  )
) )
2625, 18eqeltrd 2486 . 2  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F  e.  R )
2710, 26impbii 181 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2924    \ cdif 3285    C_ wss 3288   class class class wbr 4180    _I cid 4461   dom cdm 4845   ran crn 4846   -1-1-onto->wf1o 5420   ` cfv 5421   2oc2o 6685    ~~ cen 7073  pmTrspcpmtr 27260
This theorem is referenced by:  pmtrfconj  27283  symggen  27287  psgnunilem1  27292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-1o 6691  df-2o 6692  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pmtr 27261
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