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Theorem pmtrfcnv 27382
Description: A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t  |-  T  =  (pmTrsp `  D )
pmtrrn.r  |-  R  =  ran  T
Assertion
Ref Expression
pmtrfcnv  |-  ( F  e.  R  ->  `' F  =  F )

Proof of Theorem pmtrfcnv
StepHypRef Expression
1 pmtrrn.t . . . . . . 7  |-  T  =  (pmTrsp `  D )
2 pmtrrn.r . . . . . . 7  |-  R  =  ran  T
3 eqid 2436 . . . . . . 7  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 27377 . . . . . 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 27374 . . . . 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 5580 . . . 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 27379 . . 3  |-  ( F  e.  R  ->  ( F  o.  F )  =  (  _I  |`  D ) )
12 fcof1o 6026 . . 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 ) )
1413simprd 450 1  |-  ( F  e.  R  ->  `' F  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    \ cdif 3317    C_ wss 3320   class class class wbr 4212    _I cid 4493   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880    o. ccom 4882   -->wf 5450   -1-1-onto->wf1o 5453   ` cfv 5454   2oc2o 6718    ~~ cen 7106  pmTrspcpmtr 27361
This theorem is referenced by:  symgtrinv  27390  psgnunilem1  27393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-2o 6725  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pmtr 27362
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