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Theorem pmtrrn2 17179
Description: For any transposition there are two points it is transposing. (Contributed by SO, 15-Jul-2018.)
Hypotheses
Ref Expression
pmtrrn.t  |-  T  =  (pmTrsp `  D )
pmtrrn.r  |-  R  =  ran  T
Assertion
Ref Expression
pmtrrn2  |-  ( F  e.  R  ->  E. x  e.  D  E. y  e.  D  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) )
Distinct variable groups:    x, y, D    x, T, y    x, F, y    x, R, y

Proof of Theorem pmtrrn2
StepHypRef Expression
1 pmtrrn.t . . . . . . 7  |-  T  =  (pmTrsp `  D )
2 pmtrrn.r . . . . . . 7  |-  R  =  ran  T
3 eqid 2471 . . . . . . 7  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 17177 . . . . . 6  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
54simpld 466 . . . . 5  |-  ( F  e.  R  ->  ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o ) )
65simp3d 1044 . . . 4  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
7 en2 7825 . . . 4  |-  ( dom  ( F  \  _I  )  ~~  2o  ->  E. x E. y dom  ( F 
\  _I  )  =  { x ,  y } )
86, 7syl 17 . . 3  |-  ( F  e.  R  ->  E. x E. y dom  ( F 
\  _I  )  =  { x ,  y } )
95simp2d 1043 . . . . . . 7  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  C_  D )
104simprd 470 . . . . . . 7  |-  ( F  e.  R  ->  F  =  ( T `  dom  ( F  \  _I  ) ) )
119, 6, 10jca32 544 . . . . . 6  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  C_  D  /\  ( dom  ( F  \  _I  )  ~~  2o  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) ) )
12 sseq1 3439 . . . . . . 7  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  C_  D  <->  { x ,  y }  C_  D ) )
13 breq1 4398 . . . . . . . 8  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  ~~  2o  <->  { x ,  y }  ~~  2o ) )
14 fveq2 5879 . . . . . . . . 9  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( T `  dom  ( F  \  _I  ) )  =  ( T `  { x ,  y } ) )
1514eqeq2d 2481 . . . . . . . 8  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( F  =  ( T `
 dom  ( F  \  _I  ) )  <->  F  =  ( T `  { x ,  y } ) ) )
1613, 15anbi12d 725 . . . . . . 7  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( ( dom  ( F 
\  _I  )  ~~  2o  /\  F  =  ( T `  dom  ( F  \  _I  ) ) )  <->  ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) ) ) )
1712, 16anbi12d 725 . . . . . 6  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( ( dom  ( F 
\  _I  )  C_  D  /\  ( dom  ( F  \  _I  )  ~~  2o  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )  <->  ( {
x ,  y } 
C_  D  /\  ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) ) ) ) )
1811, 17syl5ibcom 228 . . . . 5  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( { x ,  y }  C_  D  /\  ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) ) ) ) )
19 vex 3034 . . . . . . . 8  |-  x  e. 
_V
20 vex 3034 . . . . . . . 8  |-  y  e. 
_V
2119, 20prss 4117 . . . . . . 7  |-  ( ( x  e.  D  /\  y  e.  D )  <->  { x ,  y } 
C_  D )
2221bicomi 207 . . . . . 6  |-  ( { x ,  y } 
C_  D  <->  ( x  e.  D  /\  y  e.  D ) )
23 pr2ne 8454 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( { x ,  y }  ~~  2o  <->  x  =/=  y ) )
2419, 20, 23mp2an 686 . . . . . . 7  |-  ( { x ,  y } 
~~  2o  <->  x  =/=  y
)
2524anbi1i 709 . . . . . 6  |-  ( ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) )  <->  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) )
2622, 25anbi12i 711 . . . . 5  |-  ( ( { x ,  y }  C_  D  /\  ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) ) )  <-> 
( ( x  e.  D  /\  y  e.  D )  /\  (
x  =/=  y  /\  F  =  ( T `  { x ,  y } ) ) ) )
2718, 26syl6ib 234 . . . 4  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( ( x  e.  D  /\  y  e.  D
)  /\  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) ) ) )
28272eximdv 1774 . . 3  |-  ( F  e.  R  ->  ( E. x E. y dom  ( F  \  _I  )  =  { x ,  y }  ->  E. x E. y ( ( x  e.  D  /\  y  e.  D
)  /\  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) ) ) )
298, 28mpd 15 . 2  |-  ( F  e.  R  ->  E. x E. y ( ( x  e.  D  /\  y  e.  D )  /\  (
x  =/=  y  /\  F  =  ( T `  { x ,  y } ) ) ) )
30 r2ex 2901 . 2  |-  ( E. x  e.  D  E. y  e.  D  (
x  =/=  y  /\  F  =  ( T `  { x ,  y } ) )  <->  E. x E. y ( ( x  e.  D  /\  y  e.  D )  /\  (
x  =/=  y  /\  F  =  ( T `  { x ,  y } ) ) ) )
3129, 30sylibr 217 1  |-  ( F  e.  R  ->  E. x  e.  D  E. y  e.  D  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031    \ cdif 3387    C_ wss 3390   {cpr 3961   class class class wbr 4395    _I cid 4749   dom cdm 4839   ran crn 4840   ` cfv 5589   2oc2o 7194    ~~ cen 7584  pmTrspcpmtr 17160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-2o 7201  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pmtr 17161
This theorem is referenced by:  mdetunilem7  19720
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