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Theorem pmtrfmvdn0 16070
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 7029 . 2  |-  2o  =/=  (/)
2 pmtrrn.t . . . . . . . 8  |-  T  =  (pmTrsp `  D )
3 pmtrrn.r . . . . . . . 8  |-  R  =  ran  T
4 eqid 2451 . . . . . . . 8  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
52, 3, 4pmtrfrn 16066 . . . . . . 7  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
65simpld 459 . . . . . 6  |-  ( F  e.  R  ->  ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o ) )
76simp3d 1002 . . . . 5  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
8 enen1 7551 . . . . 5  |-  ( dom  ( F  \  _I  )  ~~  2o  ->  ( dom  ( F  \  _I  )  ~~  (/)  <->  2o  ~~  (/) ) )
97, 8syl 16 . . . 4  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  ~~  (/)  <->  2o  ~~  (/) ) )
10 en0 7472 . . . 4  |-  ( dom  ( F  \  _I  )  ~~  (/)  <->  dom  ( F  \  _I  )  =  (/) )
11 en0 7472 . . . 4  |-  ( 2o 
~~  (/)  <->  2o  =  (/) )
129, 10, 113bitr3g 287 . . 3  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  (/)  <->  2o  =  (/) ) )
1312necon3bid 2706 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   _Vcvv 3068    \ cdif 3423    C_ wss 3426   (/)c0 3735   class class class wbr 4390    _I cid 4729   dom cdm 4938   ran crn 4939   ` cfv 5516   2oc2o 7014    ~~ cen 7407  pmTrspcpmtr 16049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-om 6577  df-1o 7020  df-2o 7021  df-er 7201  df-en 7411  df-fin 7414  df-pmtr 16050
This theorem is referenced by:  psgnunilem3  16104
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