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Theorem pmtrfmvdn0 27371
Description: A transpositon 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 6725 . 2  |-  2o  =/=  (/)
2 pmtrrn.t . . . . . . . 8  |-  T  =  (pmTrsp `  D )
3 pmtrrn.r . . . . . . . 8  |-  R  =  ran  T
4 eqid 2435 . . . . . . . 8  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
52, 3, 4pmtrfrn 27368 . . . . . . 7  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
65simpld 446 . . . . . 6  |-  ( F  e.  R  ->  ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o ) )
76simp3d 971 . . . . 5  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
8 enen1 7239 . . . . 5  |-  ( dom  ( F  \  _I  )  ~~  2o  ->  ( dom  ( F  \  _I  )  ~~  (/)  <->  2o  ~~  (/) ) )
97, 8syl 16 . . . 4  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  ~~  (/)  <->  2o  ~~  (/) ) )
10 en0 7162 . . . 4  |-  ( dom  ( F  \  _I  )  ~~  (/)  <->  dom  ( F  \  _I  )  =  (/) )
11 en0 7162 . . . 4  |-  ( 2o 
~~  (/)  <->  2o  =  (/) )
129, 10, 113bitr3g 279 . . 3  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  (/)  <->  2o  =  (/) ) )
1312necon3bid 2633 . 2  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =/=  (/)  <->  2o  =/=  (/) ) )
141, 13mpbiri 225 1  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   class class class wbr 4204    _I cid 4485   dom cdm 4870   ran crn 4871   ` cfv 5446   2oc2o 6710    ~~ cen 7098  pmTrspcpmtr 27352
This theorem is referenced by:  psgnunilem3  27387
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-2o 6717  df-er 6897  df-en 7102  df-fin 7105  df-pmtr 27353
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