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Theorem pmtrrn2 16359
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 2443 . . . . . . 7  |-  dom  ( F  \  _I  )  =  dom  ( F  \  _I  )
41, 2, 3pmtrfrn 16357 . . . . . 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 1011 . . . 4  |-  ( F  e.  R  ->  dom  ( F  \  _I  )  ~~  2o )
7 en2 7758 . . . 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 1010 . . . . . . 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 3510 . . . . . . 7  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  C_  D  <->  { x ,  y }  C_  D ) )
13 breq1 4440 . . . . . . . 8  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( dom  ( F  \  _I  )  ~~  2o  <->  { x ,  y }  ~~  2o ) )
14 fveq2 5856 . . . . . . . . 9  |-  ( dom  ( F  \  _I  )  =  { x ,  y }  ->  ( T `  dom  ( F  \  _I  ) )  =  ( T `  { x ,  y } ) )
1514eqeq2d 2457 . . . . . . . 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 3098 . . . . . . . 8  |-  x  e. 
_V
20 vex 3098 . . . . . . . 8  |-  y  e. 
_V
2119, 20prss 4169 . . . . . . 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 8386 . . . . . . . 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 1699 . . 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 2966 . 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 974    = wceq 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   E.wrex 2794   _Vcvv 3095    \ cdif 3458    C_ wss 3461   {cpr 4016   class class class wbr 4437    _I cid 4780   dom cdm 4989   ran crn 4990   ` cfv 5578   2oc2o 7126    ~~ cen 7515  pmTrspcpmtr 16340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-1o 7132  df-2o 7133  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pmtr 16341
This theorem is referenced by:  mdetunilem7  18993
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