Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmtrrn Unicode version

Theorem pmtrrn 27502
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 5761 . . . . . . 7  |-  ( D  e.  V  ->  (
y  e.  D  |->  if ( y  e.  z ,  U. ( z 
\  { y } ) ,  y ) )  e.  _V )
21ralrimivw 2640 . . . . . 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 976 . . . . 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 2296 . . . . . 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 5386 . . . . 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 15 . . . 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 27496 . . . . . 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 976 . . . . 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 5351 . . . 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 223 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  T  Fn  { x  e.  ~P D  |  x  ~~  2o } )
12 elpw2g 4190 . . . . . 6  |-  ( D  e.  V  ->  ( P  e.  ~P D  <->  P 
C_  D ) )
1312biimpar 471 . . . . 5  |-  ( ( D  e.  V  /\  P  C_  D )  ->  P  e.  ~P D
)
14133adant3 975 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  ~P D )
15 simp3 957 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  ~~  2o )
16 breq1 4042 . . . . 5  |-  ( x  =  P  ->  (
x  ~~  2o  <->  P  ~~  2o ) )
1716elrab 2936 . . . 4  |-  ( P  e.  { x  e. 
~P D  |  x 
~~  2o }  <->  ( P  e.  ~P D  /\  P  ~~  2o ) )
1814, 15, 17sylanbrc 645 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  { x  e.  ~P D  |  x  ~~  2o } )
19 fnfvelrn 5678 . . 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 642 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  ran  T )
21 pmtrrn.r . 2  |-  R  =  ran  T
2220, 21syl6eleqr 2387 1  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  e.  R )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   ifcif 3578   ~Pcpw 3638   {csn 3653   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   ran crn 4706    Fn wfn 5266   ` cfv 5271   2oc2o 6489    ~~ cen 6876  pmTrspcpmtr 27487
This theorem is referenced by:  pmtrfb  27509  symggen  27514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-pmtr 27488
  Copyright terms: Public domain W3C validator