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Theorem pmtrff1o 27507
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 2296 . . . . . . 7  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 27503 . . . . . 6  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
54simpld 445 . . . . 5  |-  ( F  e.  R  ->  ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o ) )
61pmtrf 27500 . . . . 5  |-  ( ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  ->  ( T `  dom  ( F 
\  _I  ) ) : D --> D )
75, 6syl 15 . . . 4  |-  ( F  e.  R  ->  ( T `  dom  ( F 
\  _I  ) ) : D --> D )
84simprd 449 . . . . 5  |-  ( F  e.  R  ->  F  =  ( T `  dom  ( F  \  _I  ) ) )
98feq1d 5395 . . . 4  |-  ( F  e.  R  ->  ( F : D --> D  <->  ( T `  dom  ( F  \  _I  ) ) : D --> D ) )
107, 9mpbird 223 . . 3  |-  ( F  e.  R  ->  F : D --> D )
111, 2pmtrfinv 27505 . . 3  |-  ( F  e.  R  ->  ( F  o.  F )  =  (  _I  |`  D ) )
12 fcof1o 5819 . . 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 1183 . 2  |-  ( F  e.  R  ->  ( F : D -1-1-onto-> D  /\  `' F  =  F ) )
1413simpld 445 1  |-  ( F  e.  R  ->  F : D -1-1-onto-> D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   class class class wbr 4039    _I cid 4320   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707    o. ccom 4709   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271   2oc2o 6489    ~~ cen 6876  pmTrspcpmtr 27487
This theorem is referenced by:  pmtrfb  27509  pmtrfconj  27510  symgtrf  27513  psgnunilem1  27519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-2o 6496  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pmtr 27488
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