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Theorem pmtrfmvdn0 16689
Description: A transposition moves at least one point. (Contributed by Stefan O'Rear, 23-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t  |-  T  =  (pmTrsp `  D )
pmtrrn.r  |-  R  =  ran  T
Assertion
Ref Expression
pmtrfmvdn0  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  =/=  (/) )

Proof of Theorem pmtrfmvdn0
StepHypRef Expression
1 2on0 7131 . 2  |-  2o  =/=  (/)
2 pmtrrn.t . . . . . . . 8  |-  T  =  (pmTrsp `  D )
3 pmtrrn.r . . . . . . . 8  |-  R  =  ran  T
4 eqid 2454 . . . . . . . 8  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
52, 3, 4pmtrfrn 16685 . . . . . . 7  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
65simpld 457 . . . . . 6  |-  ( F  e.  R  ->  ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o ) )
76simp3d 1008 . . . . 5  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
8 enen1 7650 . . . . 5  |-  ( dom  ( F  \  _I  )  ~~  2o  ->  ( dom  ( F  \  _I  )  ~~  (/)  <->  2o  ~~  (/) ) )
97, 8syl 16 . . . 4  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  ~~  (/)  <->  2o  ~~  (/) ) )
10 en0 7571 . . . 4  |-  ( dom  ( F  \  _I  )  ~~  (/)  <->  dom  ( F  \  _I  )  =  (/) )
11 en0 7571 . . . 4  |-  ( 2o 
~~  (/)  <->  2o  =  (/) )
129, 10, 113bitr3g 287 . . 3  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  (/)  <->  2o  =  (/) ) )
1312necon3bid 2712 . 2  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =/=  (/)  <->  2o  =/=  (/) ) )
141, 13mpbiri 233 1  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    \ cdif 3458    C_ wss 3461   (/)c0 3783   class class class wbr 4439    _I cid 4779   dom cdm 4988   ran crn 4989   ` cfv 5570   2oc2o 7116    ~~ cen 7506  pmTrspcpmtr 16668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-2o 7123  df-er 7303  df-en 7510  df-fin 7513  df-pmtr 16669
This theorem is referenced by:  psgnunilem3  16723
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