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Theorem pmtrfrn 15964
Description: A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t  |-  T  =  (pmTrsp `  D )
pmtrrn.r  |-  R  =  ran  T
pmtrfrn.p  |-  P  =  dom  ( F  \  _I  )
Assertion
Ref Expression
pmtrfrn  |-  ( F  e.  R  ->  (
( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) )

Proof of Theorem pmtrfrn
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3641 . . . 4  |-  -.  F  e.  (/)
2 pmtrrn.r . . . . . 6  |-  R  =  ran  T
3 pmtrrn.t . . . . . . . . 9  |-  T  =  (pmTrsp `  D )
4 fvprc 5685 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  (pmTrsp `  D )  =  (/) )
53, 4syl5eq 2487 . . . . . . . 8  |-  ( -.  D  e.  _V  ->  T  =  (/) )
65rneqd 5067 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ran 
T  =  ran  (/) )
7 rn0 5091 . . . . . . 7  |-  ran  (/)  =  (/)
86, 7syl6eq 2491 . . . . . 6  |-  ( -.  D  e.  _V  ->  ran 
T  =  (/) )
92, 8syl5eq 2487 . . . . 5  |-  ( -.  D  e.  _V  ->  R  =  (/) )
109eleq2d 2510 . . . 4  |-  ( -.  D  e.  _V  ->  ( F  e.  R  <->  F  e.  (/) ) )
111, 10mtbiri 303 . . 3  |-  ( -.  D  e.  _V  ->  -.  F  e.  R )
1211con4i 130 . 2  |-  ( F  e.  R  ->  D  e.  _V )
13 mptexg 5947 . . . . . . . 8  |-  ( D  e.  _V  ->  (
z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) )  e.  _V )
1413ralrimivw 2800 . . . . . . 7  |-  ( D  e.  _V  ->  A. w  e.  { x  e.  ~P D  |  x  ~~  2o }  ( z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) )  e.  _V )
15 eqid 2443 . . . . . . . 8  |-  ( w  e.  { x  e. 
~P D  |  x 
~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) ) )  =  ( w  e.  { x  e. 
~P D  |  x 
~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) ) )
1615fnmpt 5537 . . . . . . 7  |-  ( A. w  e.  { x  e.  ~P D  |  x 
~~  2o }  (
z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) )  e.  _V  ->  (
w  e.  { x  e.  ~P D  |  x 
~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) ) )  Fn  { x  e.  ~P D  |  x 
~~  2o } )
1714, 16syl 16 . . . . . 6  |-  ( D  e.  _V  ->  (
w  e.  { x  e.  ~P D  |  x 
~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) ) )  Fn  { x  e.  ~P D  |  x 
~~  2o } )
183pmtrfval 15956 . . . . . . 7  |-  ( D  e.  _V  ->  T  =  ( w  e. 
{ x  e.  ~P D  |  x  ~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) ) ) )
1918fneq1d 5501 . . . . . 6  |-  ( D  e.  _V  ->  ( T  Fn  { x  e.  ~P D  |  x 
~~  2o }  <->  ( w  e.  { x  e.  ~P D  |  x  ~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) ) )  Fn 
{ x  e.  ~P D  |  x  ~~  2o } ) )
2017, 19mpbird 232 . . . . 5  |-  ( D  e.  _V  ->  T  Fn  { x  e.  ~P D  |  x  ~~  2o } )
21 fvelrnb 5739 . . . . 5  |-  ( T  Fn  { x  e. 
~P D  |  x 
~~  2o }  ->  ( F  e.  ran  T  <->  E. y  e.  { x  e.  ~P D  |  x 
~~  2o }  ( T `  y )  =  F ) )
2220, 21syl 16 . . . 4  |-  ( D  e.  _V  ->  ( F  e.  ran  T  <->  E. y  e.  { x  e.  ~P D  |  x  ~~  2o }  ( T `  y )  =  F ) )
232eleq2i 2507 . . . 4  |-  ( F  e.  R  <->  F  e.  ran  T )
24 breq1 4295 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  2o  <->  y  ~~  2o ) )
2524rexrab 3123 . . . . 5  |-  ( E. y  e.  { x  e.  ~P D  |  x 
~~  2o }  ( T `  y )  =  F  <->  E. y  e.  ~P  D ( y  ~~  2o  /\  ( T `  y )  =  F ) )
2625bicomi 202 . . . 4  |-  ( E. y  e.  ~P  D
( y  ~~  2o  /\  ( T `  y
)  =  F )  <->  E. y  e.  { x  e.  ~P D  |  x 
~~  2o }  ( T `  y )  =  F )
2722, 23, 263bitr4g 288 . . 3  |-  ( D  e.  _V  ->  ( F  e.  R  <->  E. y  e.  ~P  D ( y 
~~  2o  /\  ( T `  y )  =  F ) ) )
28 elpwi 3869 . . . . 5  |-  ( y  e.  ~P D  -> 
y  C_  D )
29 simp1 988 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  D  e.  _V )
303pmtrmvd 15962 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  =  y )
31 simp2 989 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  C_  D )
3230, 31eqsstrd 3390 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  C_  D )
33 simp3 990 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  ~~  2o )
3430, 33eqbrtrd 4312 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  ~~  2o )
3529, 32, 343jca 1168 . . . . . . . . 9  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  ( D  e.  _V  /\  dom  ( ( T `  y )  \  _I  )  C_  D  /\  dom  ( ( T `  y )  \  _I  )  ~~  2o ) )
3630eqcomd 2448 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  =  dom  ( ( T `
 y )  \  _I  ) )
3736fveq2d 5695 . . . . . . . . 9  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  ( T `  y )  =  ( T `  dom  ( ( T `  y )  \  _I  ) ) )
3835, 37jca 532 . . . . . . . 8  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  (
( D  e.  _V  /\ 
dom  ( ( T `
 y )  \  _I  )  C_  D  /\  dom  ( ( T `  y )  \  _I  )  ~~  2o )  /\  ( T `  y )  =  ( T `  dom  ( ( T `  y )  \  _I  ) ) ) )
39 difeq1 3467 . . . . . . . . . . 11  |-  ( ( T `  y )  =  F  ->  (
( T `  y
)  \  _I  )  =  ( F  \  _I  ) )
4039dmeqd 5042 . . . . . . . . . 10  |-  ( ( T `  y )  =  F  ->  dom  ( ( T `  y )  \  _I  )  =  dom  ( F 
\  _I  ) )
41 pmtrfrn.p . . . . . . . . . 10  |-  P  =  dom  ( F  \  _I  )
4240, 41syl6eqr 2493 . . . . . . . . 9  |-  ( ( T `  y )  =  F  ->  dom  ( ( T `  y )  \  _I  )  =  P )
43 sseq1 3377 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( dom  ( ( T `
 y )  \  _I  )  C_  D  <->  P  C_  D
) )
44 breq1 4295 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( dom  ( ( T `
 y )  \  _I  )  ~~  2o  <->  P  ~~  2o ) )
4543, 443anbi23d 1292 . . . . . . . . . . 11  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( ( D  e.  _V  /\ 
dom  ( ( T `
 y )  \  _I  )  C_  D  /\  dom  ( ( T `  y )  \  _I  )  ~~  2o )  <->  ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o ) ) )
4645adantl 466 . . . . . . . . . 10  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( ( D  e. 
_V  /\  dom  ( ( T `  y ) 
\  _I  )  C_  D  /\  dom  ( ( T `  y ) 
\  _I  )  ~~  2o )  <->  ( D  e. 
_V  /\  P  C_  D  /\  P  ~~  2o ) ) )
47 simpl 457 . . . . . . . . . . 11  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( T `  y
)  =  F )
48 fveq2 5691 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( T `  dom  (
( T `  y
)  \  _I  )
)  =  ( T `
 P ) )
4948adantl 466 . . . . . . . . . . 11  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( T `  dom  ( ( T `  y )  \  _I  ) )  =  ( T `  P ) )
5047, 49eqeq12d 2457 . . . . . . . . . 10  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( ( T `  y )  =  ( T `  dom  (
( T `  y
)  \  _I  )
)  <->  F  =  ( T `  P )
) )
5146, 50anbi12d 710 . . . . . . . . 9  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( ( ( D  e.  _V  /\  dom  ( ( T `  y )  \  _I  )  C_  D  /\  dom  ( ( T `  y )  \  _I  )  ~~  2o )  /\  ( T `  y )  =  ( T `  dom  ( ( T `  y )  \  _I  ) ) )  <->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P
) ) ) )
5242, 51mpdan 668 . . . . . . . 8  |-  ( ( T `  y )  =  F  ->  (
( ( D  e. 
_V  /\  dom  ( ( T `  y ) 
\  _I  )  C_  D  /\  dom  ( ( T `  y ) 
\  _I  )  ~~  2o )  /\  ( T `  y )  =  ( T `  dom  ( ( T `  y )  \  _I  ) ) )  <->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P
) ) ) )
5338, 52syl5ibcom 220 . . . . . . 7  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  (
( T `  y
)  =  F  -> 
( ( D  e. 
_V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) )
54533exp 1186 . . . . . 6  |-  ( D  e.  _V  ->  (
y  C_  D  ->  ( y  ~~  2o  ->  ( ( T `  y
)  =  F  -> 
( ( D  e. 
_V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) ) ) )
5554imp4a 589 . . . . 5  |-  ( D  e.  _V  ->  (
y  C_  D  ->  ( ( y  ~~  2o  /\  ( T `  y
)  =  F )  ->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P
) ) ) ) )
5628, 55syl5 32 . . . 4  |-  ( D  e.  _V  ->  (
y  e.  ~P D  ->  ( ( y  ~~  2o  /\  ( T `  y )  =  F )  ->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P
) ) ) ) )
5756rexlimdv 2840 . . 3  |-  ( D  e.  _V  ->  ( E. y  e.  ~P  D ( y  ~~  2o  /\  ( T `  y )  =  F )  ->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P
) ) ) )
5827, 57sylbid 215 . 2  |-  ( D  e.  _V  ->  ( F  e.  R  ->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) )
5912, 58mpcom 36 1  |-  ( F  e.  R  ->  (
( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972    \ cdif 3325    C_ wss 3328   (/)c0 3637   ifcif 3791   ~Pcpw 3860   {csn 3877   U.cuni 4091   class class class wbr 4292    e. cmpt 4350    _I cid 4631   dom cdm 4840   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-8 1758  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  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  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-om 6477  df-1o 6920  df-2o 6921  df-er 7101  df-en 7311  df-fin 7314  df-pmtr 15948
This theorem is referenced by:  pmtrffv  15965  pmtrrn2  15966  pmtrfinv  15967  pmtrfmvdn0  15968  pmtrff1o  15969  pmtrfcnv  15970  pmtrfb  15971
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