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Theorem pmtrff1o 27066
Description: A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t  |-  T  =  (pmTrsp `  D )
pmtrrn.r  |-  R  =  ran  T
Assertion
Ref Expression
pmtrff1o  |-  ( F  e.  R  ->  F : D -1-1-onto-> D )

Proof of Theorem pmtrff1o
StepHypRef Expression
1 pmtrrn.t . . . . . . 7  |-  T  =  (pmTrsp `  D )
2 pmtrrn.r . . . . . . 7  |-  R  =  ran  T
3 eqid 2380 . . . . . . 7  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 27062 . . . . . 6  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
54simpld 446 . . . . 5  |-  ( F  e.  R  ->  ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o ) )
61pmtrf 27059 . . . . 5  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `  dom  ( F 
\  _I  ) ) : D --> D )
75, 6syl 16 . . . 4  |-  ( F  e.  R  ->  ( T `  dom  ( F 
\  _I  ) ) : D --> D )
84simprd 450 . . . . 5  |-  ( F  e.  R  ->  F  =  ( T `  dom  ( F  \  _I  ) ) )
98feq1d 5513 . . . 4  |-  ( F  e.  R  ->  ( F : D --> D  <->  ( T `  dom  ( F  \  _I  ) ) : D --> D ) )
107, 9mpbird 224 . . 3  |-  ( F  e.  R  ->  F : D --> D )
111, 2pmtrfinv 27064 . . 3  |-  ( F  e.  R  ->  ( F  o.  F )  =  (  _I  |`  D ) )
12 fcof1o 5958 . . 3  |-  ( ( ( F : D --> D  /\  F : D --> D )  /\  (
( F  o.  F
)  =  (  _I  |`  D )  /\  ( F  o.  F )  =  (  _I  |`  D ) ) )  ->  ( F : D -1-1-onto-> D  /\  `' F  =  F ) )
1310, 10, 11, 11, 12syl22anc 1185 . 2  |-  ( F  e.  R  ->  ( F : D -1-1-onto-> D  /\  `' F  =  F ) )
1413simpld 446 1  |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2892    \ cdif 3253    C_ wss 3256   class class class wbr 4146    _I cid 4427   `'ccnv 4810   dom cdm 4811   ran crn 4812    |` cres 4813    o. ccom 4815   -->wf 5383   -1-1-onto->wf1o 5386   ` cfv 5387   2oc2o 6647    ~~ cen 7035  pmTrspcpmtr 27046
This theorem is referenced by:  pmtrfb  27068  pmtrfconj  27069  symgtrf  27072  psgnunilem1  27078
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-1o 6653  df-2o 6654  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pmtr 27047
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