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Theorem pmtrfb 15986
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 2443 . . . . 5  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 15979 . . . 4  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
5 simpl1 991 . . . 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 15984 . . 3  |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
8 simpl3 993 . . . 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 1168 . 2  |-  ( F  e.  R  ->  ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o ) )
11 simp2 989 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F : D
-1-1-onto-> D )
12 difss 3498 . . . . . . . 8  |-  ( F 
\  _I  )  C_  F
13 dmss 5054 . . . . . . . 8  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
1412, 13ax-mp 5 . . . . . . 7  |-  dom  ( F  \  _I  )  C_  dom  F
15 f1odm 5660 . . . . . . 7  |-  ( F : D -1-1-onto-> D  ->  dom  F  =  D )
1614, 15syl5sseq 3419 . . . . . 6  |-  ( F : D -1-1-onto-> D  ->  dom  ( F 
\  _I  )  C_  D )
171, 2pmtrrn 15978 . . . . . 6  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `  dom  ( F 
\  _I  ) )  e.  R )
1816, 17syl3an2 1252 . . . . 5  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `
 dom  ( F  \  _I  ) )  e.  R )
191, 2pmtrff1o 15984 . . . . 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 990 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  ( F  \  _I  )  ~~  2o )
221pmtrmvd 15977 . . . . 5  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  ( ( T `  dom  ( F  \  _I  ) )  \  _I  )  =  dom  ( F 
\  _I  ) )
2316, 22syl3an2 1252 . . . 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 15968 . . . 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 1219 . . 3  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F  =  ( T `  dom  ( F  \  _I  )
) )
2625, 18eqeltrd 2517 . 2  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F  e.  R )
2710, 26impbii 188 1  |-  ( F  e.  R  <->  ( D  e.  _V  /\  F : D
-1-1-onto-> D  /\  dom  ( F 
\  _I  )  ~~  2o ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2987    \ cdif 3340    C_ wss 3343   class class class wbr 4307    _I cid 4646   dom cdm 4855   ran crn 4856   -1-1-onto->wf1o 5432   ` cfv 5433   2oc2o 6929    ~~ cen 7322  pmTrspcpmtr 15962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-om 6492  df-1o 6935  df-2o 6936  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-pmtr 15963
This theorem is referenced by:  pmtrfconj  15987  symggen  15991  pmtrdifellem1  15997  pmtrdifellem2  15998  psgnunilem1  16014
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