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Theorem pmtrfrn 27400
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 3459 . . . 4  |-  -.  F  e.  (/)
2 pmtrrn.r . . . . . 6  |-  R  =  ran  T
3 pmtrrn.t . . . . . . . . 9  |-  T  =  (pmTrsp `  D )
4 fvprc 5519 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  (pmTrsp `  D )  =  (/) )
53, 4syl5eq 2327 . . . . . . . 8  |-  ( -.  D  e.  _V  ->  T  =  (/) )
65rneqd 4906 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ran 
T  =  ran  (/) )
7 rn0 4936 . . . . . . 7  |-  ran  (/)  =  (/)
86, 7syl6eq 2331 . . . . . 6  |-  ( -.  D  e.  _V  ->  ran 
T  =  (/) )
92, 8syl5eq 2327 . . . . 5  |-  ( -.  D  e.  _V  ->  R  =  (/) )
109eleq2d 2350 . . . 4  |-  ( -.  D  e.  _V  ->  ( F  e.  R  <->  F  e.  (/) ) )
111, 10mtbiri 294 . . 3  |-  ( -.  D  e.  _V  ->  -.  F  e.  R )
1211con4i 122 . 2  |-  ( F  e.  R  ->  D  e.  _V )
13 mptexg 5745 . . . . . . . 8  |-  ( D  e.  _V  ->  (
z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) )  e.  _V )
1413ralrimivw 2627 . . . . . . 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 2283 . . . . . . . 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 5370 . . . . . . 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 15 . . . . . 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 27393 . . . . . . 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 5335 . . . . . 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 223 . . . . 5  |-  ( D  e.  _V  ->  T  Fn  { x  e.  ~P D  |  x  ~~  2o } )
21 fvelrnb 5570 . . . . 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 15 . . . 4  |-  ( D  e.  _V  ->  ( F  e.  ran  T  <->  E. y  e.  { x  e.  ~P D  |  x  ~~  2o }  ( T `  y )  =  F ) )
232eleq2i 2347 . . . 4  |-  ( F  e.  R  <->  F  e.  ran  T )
24 breq1 4026 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  2o  <->  y  ~~  2o ) )
2524rexrab 2929 . . . . 5  |-  ( E. y  e.  { x  e.  ~P D  |  x 
~~  2o }  ( T `  y )  =  F  <->  E. y  e.  ~P  D ( y  ~~  2o  /\  ( T `  y )  =  F ) )
2625bicomi 193 . . . 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 279 . . 3  |-  ( D  e.  _V  ->  ( F  e.  R  <->  E. y  e.  ~P  D ( y 
~~  2o  /\  ( T `  y )  =  F ) ) )
28 elpwi 3633 . . . . 5  |-  ( y  e.  ~P D  -> 
y  C_  D )
29 simp1 955 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  D  e.  _V )
303pmtrmvd 27398 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  =  y )
31 simp2 956 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  C_  D )
3230, 31eqsstrd 3212 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  C_  D )
33 simp3 957 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  ~~  2o )
3430, 33eqbrtrd 4043 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  ~~  2o )
3529, 32, 343jca 1132 . . . . . . . . 9  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  ( D  e.  _V  /\  dom  ( ( T `  y )  \  _I  )  C_  D  /\  dom  ( ( T `  y )  \  _I  )  ~~  2o ) )
3630eqcomd 2288 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  =  dom  ( ( T `
 y )  \  _I  ) )
3736fveq2d 5529 . . . . . . . . 9  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  ( T `  y )  =  ( T `  dom  ( ( T `  y )  \  _I  ) ) )
3835, 37jca 518 . . . . . . . 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 3287 . . . . . . . . . . 11  |-  ( ( T `  y )  =  F  ->  (
( T `  y
)  \  _I  )  =  ( F  \  _I  ) )
4039dmeqd 4881 . . . . . . . . . 10  |-  ( ( T `  y )  =  F  ->  dom  ( ( T `  y )  \  _I  )  =  dom  ( F 
\  _I  ) )
41 pmtrfrn.p . . . . . . . . . 10  |-  P  =  dom  ( F  \  _I  )
4240, 41syl6eqr 2333 . . . . . . . . 9  |-  ( ( T `  y )  =  F  ->  dom  ( ( T `  y )  \  _I  )  =  P )
43 sseq1 3199 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( dom  ( ( T `
 y )  \  _I  )  C_  D  <->  P  C_  D
) )
44 breq1 4026 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( dom  ( ( T `
 y )  \  _I  )  ~~  2o  <->  P  ~~  2o ) )
4543, 443anbi23d 1255 . . . . . . . . . . 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 452 . . . . . . . . . 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 443 . . . . . . . . . . 11  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( T `  y
)  =  F )
48 fveq2 5525 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( T `  dom  (
( T `  y
)  \  _I  )
)  =  ( T `
 P ) )
4948adantl 452 . . . . . . . . . . 11  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( T `  dom  ( ( T `  y )  \  _I  ) )  =  ( T `  P ) )
5047, 49eqeq12d 2297 . . . . . . . . . 10  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( ( T `  y )  =  ( T `  dom  (
( T `  y
)  \  _I  )
)  <->  F  =  ( T `  P )
) )
5146, 50anbi12d 691 . . . . . . . . 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 649 . . . . . . . 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 211 . . . . . . 7  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  (
( T `  y
)  =  F  -> 
( ( D  e. 
_V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) )
54533exp 1150 . . . . . 6  |-  ( D  e.  _V  ->  (
y  C_  D  ->  ( y  ~~  2o  ->  ( ( T `  y
)  =  F  -> 
( ( D  e. 
_V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) ) ) )
5554imp4a 572 . . . . 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 28 . . . 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 2666 . . 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 206 . 2  |-  ( D  e.  _V  ->  ( F  e.  R  ->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) )
5912, 58mpcom 32 1  |-  ( F  e.  R  ->  (
( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   ifcif 3565   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    _I cid 4304   dom cdm 4689   ran crn 4690    Fn wfn 5250   ` cfv 5255   2oc2o 6473    ~~ cen 6860  pmTrspcpmtr 27384
This theorem is referenced by:  pmtrffv  27401  pmtrfinv  27402  pmtrfmvdn0  27403  pmtrff1o  27404  pmtrfcnv  27405  pmtrfb  27406
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-fin 6867  df-pmtr 27385
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