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Theorem pmtrrn 27267
Description: Transposing two points gives a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t  |-  T  =  (pmTrsp `  D )
pmtrrn.r  |-  R  =  ran  T
Assertion
Ref Expression
pmtrrn  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  R )

Proof of Theorem pmtrrn
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mptexg 5924 . . . . . . 7  |-  ( D  e.  V  ->  (
y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) )  e.  _V )
21ralrimivw 2750 . . . . . 6  |-  ( D  e.  V  ->  A. z  e.  { x  e.  ~P D  |  x  ~~  2o }  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z  \  {
y } ) ,  y ) )  e. 
_V )
323ad2ant1 978 . . . . 5  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  A. z  e.  { x  e.  ~P D  |  x  ~~  2o }  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z  \  {
y } ) ,  y ) )  e. 
_V )
4 eqid 2404 . . . . . 6  |-  ( z  e.  { x  e. 
~P D  |  x 
~~  2o }  |->  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) ) )  =  ( z  e.  { x  e.  ~P D  |  x 
~~  2o }  |->  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) ) )
54fnmpt 5530 . . . . 5  |-  ( A. z  e.  { x  e.  ~P D  |  x 
~~  2o }  (
y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) )  e.  _V  ->  ( z  e.  { x  e.  ~P D  |  x 
~~  2o }  |->  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) ) )  Fn  {
x  e.  ~P D  |  x  ~~  2o }
)
63, 5syl 16 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
z  e.  { x  e.  ~P D  |  x 
~~  2o }  |->  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) ) )  Fn  {
x  e.  ~P D  |  x  ~~  2o }
)
7 pmtrrn.t . . . . . . 7  |-  T  =  (pmTrsp `  D )
87pmtrfval 27261 . . . . . 6  |-  ( D  e.  V  ->  T  =  ( z  e. 
{ x  e.  ~P D  |  x  ~~  2o }  |->  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z  \  {
y } ) ,  y ) ) ) )
983ad2ant1 978 . . . . 5  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  T  =  ( z  e. 
{ x  e.  ~P D  |  x  ~~  2o }  |->  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z  \  {
y } ) ,  y ) ) ) )
109fneq1d 5495 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T  Fn  { x  e.  ~P D  |  x 
~~  2o }  <->  ( z  e.  { x  e.  ~P D  |  x  ~~  2o }  |->  ( y  e.  D  |->  if ( y  e.  z ,  U. ( z  \  {
y } ) ,  y ) ) )  Fn  { x  e. 
~P D  |  x 
~~  2o } ) )
116, 10mpbird 224 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  T  Fn  { x  e.  ~P D  |  x  ~~  2o } )
12 elpw2g 4323 . . . . . 6  |-  ( D  e.  V  ->  ( P  e.  ~P D  <->  P 
C_  D ) )
1312biimpar 472 . . . . 5  |-  ( ( D  e.  V  /\  P  C_  D )  ->  P  e.  ~P D
)
14133adant3 977 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  ~P D )
15 simp3 959 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  ~~  2o )
16 breq1 4175 . . . . 5  |-  ( x  =  P  ->  (
x  ~~  2o  <->  P  ~~  2o ) )
1716elrab 3052 . . . 4  |-  ( P  e.  { x  e. 
~P D  |  x 
~~  2o }  <->  ( P  e.  ~P D  /\  P  ~~  2o ) )
1814, 15, 17sylanbrc 646 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  { x  e.  ~P D  |  x  ~~  2o } )
19 fnfvelrn 5826 . . 3  |-  ( ( T  Fn  { x  e.  ~P D  |  x 
~~  2o }  /\  P  e.  { x  e.  ~P D  |  x 
~~  2o } )  ->  ( T `  P )  e.  ran  T )
2011, 18, 19syl2anc 643 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  ran  T )
21 pmtrrn.r . 2  |-  R  =  ran  T
2220, 21syl6eleqr 2495 1  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670   _Vcvv 2916    \ cdif 3277    C_ wss 3280   ifcif 3699   ~Pcpw 3759   {csn 3774   U.cuni 3975   class class class wbr 4172    e. cmpt 4226   ran crn 4838    Fn wfn 5408   ` cfv 5413   2oc2o 6677    ~~ cen 7065  pmTrspcpmtr 27252
This theorem is referenced by:  pmtrfb  27274  symggen  27279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-pmtr 27253
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