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Theorem pmtrrn 27376
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 5965 . . . . . . 7  |-  ( D  e.  V  ->  (
y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) )  e.  _V )
21ralrimivw 2790 . . . . . 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 2436 . . . . . 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 5571 . . . . 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 27370 . . . . . 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 5536 . . . 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 4363 . . . . . 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 4215 . . . . 5  |-  ( x  =  P  ->  (
x  ~~  2o  <->  P  ~~  2o ) )
1716elrab 3092 . . . 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 5867 . . 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 2527 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 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    \ cdif 3317    C_ wss 3320   ifcif 3739   ~Pcpw 3799   {csn 3814   U.cuni 4015   class class class wbr 4212    e. cmpt 4266   ran crn 4879    Fn wfn 5449   ` cfv 5454   2oc2o 6718    ~~ cen 7106  pmTrspcpmtr 27361
This theorem is referenced by:  pmtrfb  27383  symggen  27388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-pmtr 27362
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