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Theorem pmtrrn2 15965
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 2442 . . . . . . 7  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 15963 . . . . . 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 1002 . . . 4  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
7 en2 7547 . . . 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 1001 . . . . . . 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 3376 . . . . . . 7  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  C_  D  <->  { x ,  y }  C_  D ) )
13 breq1 4294 . . . . . . . 8  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  ~~  2o  <->  { x ,  y }  ~~  2o ) )
14 fveq2 5690 . . . . . . . . 9  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( T `  dom  ( F  \  _I  ) )  =  ( T `  { x ,  y } ) )
1514eqeq2d 2453 . . . . . . . 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 2974 . . . . . . . 8  |-  x  e. 
_V
20 vex 2974 . . . . . . . 8  |-  y  e. 
_V
2119, 20prss 4026 . . . . . . 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 8171 . . . . . . . 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 1678 . . 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 2752 . 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 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2605   E.wrex 2715   _Vcvv 2971    \ cdif 3324    C_ wss 3327   {cpr 3878   class class class wbr 4291    _I cid 4630   dom cdm 4839   ran crn 4840   ` cfv 5417   2oc2o 6913    ~~ cen 7306  pmTrspcpmtr 15946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-om 6476  df-1o 6919  df-2o 6920  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pmtr 15947
This theorem is referenced by:  mdetunilem7  18423
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