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Theorem pmtrrn2 16291
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 2467 . . . . . . 7  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 16289 . . . . . 6  |-  ( F  e.  R  ->  (
( D  e.  _V  /\ 
dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o )  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) )
54simpld 459 . . . . 5  |-  ( F  e.  R  ->  ( D  e.  _V  /\  dom  ( F  \  _I  )  C_  D  /\  dom  ( F  \  _I  )  ~~  2o ) )
65simp3d 1010 . . . 4  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
7 en2 7756 . . . 4  |-  ( dom  ( F  \  _I  )  ~~  2o  ->  E. x E. y dom  ( F 
\  _I  )  =  { x ,  y } )
86, 7syl 16 . . 3  |-  ( F  e.  R  ->  E. x E. y dom  ( F 
\  _I  )  =  { x ,  y } )
95simp2d 1009 . . . . . . 7  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  C_  D )
104simprd 463 . . . . . . 7  |-  ( F  e.  R  ->  F  =  ( T `  dom  ( F  \  _I  ) ) )
119, 6, 10jca32 535 . . . . . 6  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  C_  D  /\  ( dom  ( F  \  _I  )  ~~  2o  /\  F  =  ( T `  dom  ( F  \  _I  ) ) ) ) )
12 sseq1 3525 . . . . . . 7  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  C_  D  <->  { x ,  y }  C_  D ) )
13 breq1 4450 . . . . . . . 8  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  ~~  2o  <->  { x ,  y }  ~~  2o ) )
14 fveq2 5866 . . . . . . . . 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 710 . . . . . . 7  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( ( dom  ( F 
\  _I  )  ~~  2o  /\  F  =  ( T `  dom  ( F  \  _I  ) ) )  <->  ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) ) ) )
1712, 16anbi12d 710 . . . . . 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 220 . . . . 5  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( { x ,  y }  C_  D  /\  ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) ) ) ) )
19 vex 3116 . . . . . . . 8  |-  x  e. 
_V
20 vex 3116 . . . . . . . 8  |-  y  e. 
_V
2119, 20prss 4181 . . . . . . 7  |-  ( ( x  e.  D  /\  y  e.  D )  <->  { x ,  y } 
C_  D )
2221bicomi 202 . . . . . 6  |-  ( { x ,  y } 
C_  D  <->  ( x  e.  D  /\  y  e.  D ) )
23 pr2ne 8383 . . . . . . . 8  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( { x ,  y }  ~~  2o  <->  x  =/=  y ) )
2419, 20, 23mp2an 672 . . . . . . 7  |-  ( { x ,  y } 
~~  2o  <->  x  =/=  y
)
2524anbi1i 695 . . . . . 6  |-  ( ( { x ,  y }  ~~  2o  /\  F  =  ( T `  { x ,  y } ) )  <->  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) )
2622, 25anbi12i 697 . . . . 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 226 . . . 4  |-  ( F  e.  R  ->  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( ( x  e.  D  /\  y  e.  D
)  /\  ( x  =/=  y  /\  F  =  ( T `  {
x ,  y } ) ) ) ) )
28272eximdv 1688 . . 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 2985 . 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 212 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 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2815   _Vcvv 3113    \ cdif 3473    C_ wss 3476   {cpr 4029   class class class wbr 4447    _I cid 4790   dom cdm 4999   ran crn 5000   ` cfv 5588   2oc2o 7124    ~~ cen 7513  pmTrspcpmtr 16272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-1o 7130  df-2o 7131  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pmtr 16273
This theorem is referenced by:  mdetunilem7  18915
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