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Theorem pmtrrn2 17101
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 2451 . . . . . . 7  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 17099 . . . . . 6  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
54simpld 461 . . . . 5  |-  ( F  e.  R  ->  ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o ) )
65simp3d 1022 . . . 4  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
7 en2 7807 . . . 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 1021 . . . . . . 7  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  C_  D )
104simprd 465 . . . . . . 7  |-  ( F  e.  R  ->  F  =  ( T `  dom  ( F  \  _I  ) ) )
119, 6, 10jca32 538 . . . . . 6  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  C_  D  /\  ( dom  ( F  \  _I  )  ~~  2o  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) ) )
12 sseq1 3453 . . . . . . 7  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  C_  D  <->  { x ,  y }  C_  D ) )
13 breq1 4405 . . . . . . . 8  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  ~~  2o  <->  { x ,  y }  ~~  2o ) )
14 fveq2 5865 . . . . . . . . 9  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( T `  dom  ( F  \  _I  ) )  =  ( T `  { x ,  y } ) )
1514eqeq2d 2461 . . . . . . . 8  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( F  =  ( T `
 dom  ( F  \  _I  ) )  <->  F  =  ( T `  { x ,  y } ) ) )
1613, 15anbi12d 717 . . . . . . 7  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( ( dom  ( F 
\  _I  )  ~~  2o  /\  F  =  ( T `  dom  ( F  \  _I  ) ) )  <->  ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) ) ) )
1712, 16anbi12d 717 . . . . . 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 224 . . . . 5  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( { x ,  y }  C_  D  /\  ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) ) ) ) )
19 vex 3048 . . . . . . . 8  |-  x  e. 
_V
20 vex 3048 . . . . . . . 8  |-  y  e. 
_V
2119, 20prss 4126 . . . . . . 7  |-  ( ( x  e.  D  /\  y  e.  D )  <->  { x ,  y } 
C_  D )
2221bicomi 206 . . . . . 6  |-  ( { x ,  y } 
C_  D  <->  ( x  e.  D  /\  y  e.  D ) )
23 pr2ne 8436 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( { x ,  y }  ~~  2o  <->  x  =/=  y ) )
2419, 20, 23mp2an 678 . . . . . . 7  |-  ( { x ,  y } 
~~  2o  <->  x  =/=  y
)
2524anbi1i 701 . . . . . 6  |-  ( ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) )  <->  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) )
2622, 25anbi12i 703 . . . . 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 230 . . . 4  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( ( x  e.  D  /\  y  e.  D
)  /\  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) ) ) )
28272eximdv 1766 . . 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 2913 . 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 216 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 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   E.wrex 2738   _Vcvv 3045    \ cdif 3401    C_ wss 3404   {cpr 3970   class class class wbr 4402    _I cid 4744   dom cdm 4834   ran crn 4835   ` cfv 5582   2oc2o 7176    ~~ cen 7566  pmTrspcpmtr 17082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-2o 7183  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pmtr 17083
This theorem is referenced by:  mdetunilem7  19643
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