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Theorem pmtrfb 26729
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 2358 . . . . 5  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 26723 . . . 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 26727 . . 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 3379 . . . . . . . 8  |-  ( F 
\  _I  )  C_  F
13 dmss 4957 . . . . . . . 8  |-  ( ( F  \  _I  )  C_  F  ->  dom  ( F 
\  _I  )  C_  dom  F )
1412, 13ax-mp 8 . . . . . . 7  |-  dom  ( F  \  _I  )  C_  dom  F
15 f1odm 5556 . . . . . . 7  |-  ( F : D -1-1-onto-> D  ->  dom  F  =  D )
1614, 15syl5sseq 3302 . . . . . 6  |-  ( F : D -1-1-onto-> D  ->  dom  ( F 
\  _I  )  C_  D )
171, 2pmtrrn 26722 . . . . . 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 26727 . . . . 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 26721 . . . . 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 26713 . . . 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 2432 . 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 1642    e. wcel 1710   _Vcvv 2864    \ cdif 3225    C_ wss 3228   class class class wbr 4102    _I cid 4383   dom cdm 4768   ran crn 4769   -1-1-onto->wf1o 5333   ` cfv 5334   2oc2o 6557    ~~ cen 6945  pmTrspcpmtr 26707
This theorem is referenced by:  pmtrfconj  26730  symggen  26734  psgnunilem1  26739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-1o 6563  df-2o 6564  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-pmtr 26708
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