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Theorem pmtrfb 27397
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 2438 . . . . 5  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 27391 . . . 4  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
5 simpl1 961 . . . 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 27395 . . 3  |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
8 simpl3 963 . . . 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 1135 . 2  |-  ( F  e.  R  ->  ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o ) )
11 simp2 959 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F : D
-1-1-onto-> D )
12 difss 3476 . . . . . . . 8  |-  ( F 
\  _I  )  C_  F
13 dmss 5072 . . . . . . . 8  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
1412, 13ax-mp 5 . . . . . . 7  |-  dom  ( F  \  _I  )  C_  dom  F
15 f1odm 5681 . . . . . . 7  |-  ( F : D -1-1-onto-> D  ->  dom  F  =  D )
1614, 15syl5sseq 3398 . . . . . 6  |-  ( F : D -1-1-onto-> D  ->  dom  ( F 
\  _I  )  C_  D )
171, 2pmtrrn 27390 . . . . . 6  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `  dom  ( F 
\  _I  ) )  e.  R )
1816, 17syl3an2 1219 . . . . 5  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `
 dom  ( F  \  _I  ) )  e.  R )
191, 2pmtrff1o 27395 . . . . 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 960 . . . 4  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  ( F  \  _I  )  ~~  2o )
221pmtrmvd 27389 . . . . 5  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  dom  ( ( T `  dom  ( F  \  _I  ) )  \  _I  )  =  dom  ( F 
\  _I  ) )
2316, 22syl3an2 1219 . . . 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 27381 . . . 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 1186 . . 3  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F  =  ( T `  dom  ( F  \  _I  )
) )
2625, 18eqeltrd 2512 . 2  |-  ( ( D  e.  _V  /\  F : D -1-1-onto-> D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  F  e.  R )
2710, 26impbii 182 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    C_ wss 3322   class class class wbr 4215    _I cid 4496   dom cdm 4881   ran crn 4882   -1-1-onto->wf1o 5456   ` cfv 5457   2oc2o 6721    ~~ cen 7109  pmTrspcpmtr 27375
This theorem is referenced by:  pmtrfconj  27398  symggen  27402  psgnunilem1  27407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-1o 6727  df-2o 6728  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pmtr 27376
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