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Theorem pmtrrn 15963
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 5947 . . . . . . 7  |-  ( D  e.  V  ->  (
y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) )  e.  _V )
21ralrimivw 2800 . . . . . 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 1009 . . . . 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 2443 . . . . . 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 5537 . . . . 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 15956 . . . . . 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 1009 . . . . 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 5501 . . . 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 232 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  T  Fn  { x  e.  ~P D  |  x  ~~  2o } )
12 elpw2g 4455 . . . . . 6  |-  ( D  e.  V  ->  ( P  e.  ~P D  <->  P 
C_  D ) )
1312biimpar 485 . . . . 5  |-  ( ( D  e.  V  /\  P  C_  D )  ->  P  e.  ~P D
)
14133adant3 1008 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  ~P D )
15 simp3 990 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  ~~  2o )
16 breq1 4295 . . . . 5  |-  ( x  =  P  ->  (
x  ~~  2o  <->  P  ~~  2o ) )
1716elrab 3117 . . . 4  |-  ( P  e.  { x  e. 
~P D  |  x 
~~  2o }  <->  ( P  e.  ~P D  /\  P  ~~  2o ) )
1814, 15, 17sylanbrc 664 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  { x  e.  ~P D  |  x  ~~  2o } )
19 fnfvelrn 5840 . . 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 661 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  ran  T )
21 pmtrrn.r . 2  |-  R  =  ran  T
2220, 21syl6eleqr 2534 1  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   _Vcvv 2972    \ cdif 3325    C_ wss 3328   ifcif 3791   ~Pcpw 3860   {csn 3877   U.cuni 4091   class class class wbr 4292    e. cmpt 4350   ran crn 4841    Fn wfn 5413   ` cfv 5418   2oc2o 6914    ~~ cen 7307  pmTrspcpmtr 15947
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-pmtr 15948
This theorem is referenced by:  pmtrfb  15971  symggen  15976  pmtr3ncom  15981  pmtrdifellem1  15982  mdetralt  18414
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