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Theorem pmtrrn 17049
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 6150 . . . . . . 7  |-  ( D  e.  V  ->  (
y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) )  e.  _V )
21ralrimivw 2847 . . . . . 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 1026 . . . . 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 2429 . . . . . 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 5722 . . . . 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 17 . . . 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 17042 . . . . . 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 1026 . . . . 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 5684 . . . 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 235 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  T  Fn  { x  e.  ~P D  |  x  ~~  2o } )
12 elpw2g 4588 . . . . . 6  |-  ( D  e.  V  ->  ( P  e.  ~P D  <->  P 
C_  D ) )
1312biimpar 487 . . . . 5  |-  ( ( D  e.  V  /\  P  C_  D )  ->  P  e.  ~P D
)
14133adant3 1025 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  ~P D )
15 simp3 1007 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  ~~  2o )
16 breq1 4429 . . . . 5  |-  ( x  =  P  ->  (
x  ~~  2o  <->  P  ~~  2o ) )
1716elrab 3235 . . . 4  |-  ( P  e.  { x  e. 
~P D  |  x 
~~  2o }  <->  ( P  e.  ~P D  /\  P  ~~  2o ) )
1814, 15, 17sylanbrc 668 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  { x  e.  ~P D  |  x  ~~  2o } )
19 fnfvelrn 6034 . . 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 665 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  ran  T )
21 pmtrrn.r . 2  |-  R  =  ran  T
2220, 21syl6eleqr 2528 1  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  R )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   {crab 2786   _Vcvv 3087    \ cdif 3439    C_ wss 3442   ifcif 3915   ~Pcpw 3985   {csn 4002   U.cuni 4222   class class class wbr 4426    |-> cmpt 4484   ran crn 4855    Fn wfn 5596   ` cfv 5601   2oc2o 7184    ~~ cen 7574  pmTrspcpmtr 17033
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-pmtr 17034
This theorem is referenced by:  pmtrfb  17057  symggen  17062  pmtr3ncom  17067  pmtrdifellem1  17068  mdetralt  19564  pmtrto1cl  28451
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