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Theorem pmtrfrn 16286
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 3789 . . . 4  |-  -.  F  e.  (/)
2 pmtrrn.r . . . . . 6  |-  R  =  ran  T
3 pmtrrn.t . . . . . . . . 9  |-  T  =  (pmTrsp `  D )
4 fvprc 5859 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  (pmTrsp `  D )  =  (/) )
53, 4syl5eq 2520 . . . . . . . 8  |-  ( -.  D  e.  _V  ->  T  =  (/) )
65rneqd 5229 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ran 
T  =  ran  (/) )
7 rn0 5253 . . . . . . 7  |-  ran  (/)  =  (/)
86, 7syl6eq 2524 . . . . . 6  |-  ( -.  D  e.  _V  ->  ran 
T  =  (/) )
92, 8syl5eq 2520 . . . . 5  |-  ( -.  D  e.  _V  ->  R  =  (/) )
109eleq2d 2537 . . . 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 6129 . . . . . . . 8  |-  ( D  e.  _V  ->  (
z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) )  e.  _V )
1413ralrimivw 2879 . . . . . . 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 2467 . . . . . . . 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 5706 . . . . . . 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 16278 . . . . . . 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 5670 . . . . . 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 5914 . . . . 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 2545 . . . 4  |-  ( F  e.  R  <->  F  e.  ran  T )
24 breq1 4450 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  2o  <->  y  ~~  2o ) )
2524rexrab 3267 . . . . 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 4019 . . . . 5  |-  ( y  e.  ~P D  -> 
y  C_  D )
29 simp1 996 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  D  e.  _V )
303pmtrmvd 16284 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  =  y )
31 simp2 997 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  C_  D )
3230, 31eqsstrd 3538 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  C_  D )
33 simp3 998 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  ~~  2o )
3430, 33eqbrtrd 4467 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  ~~  2o )
3529, 32, 343jca 1176 . . . . . . . . 9  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  ( D  e.  _V  /\  dom  ( ( T `  y )  \  _I  )  C_  D  /\  dom  ( ( T `  y )  \  _I  )  ~~  2o ) )
3630eqcomd 2475 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  =  dom  ( ( T `
 y )  \  _I  ) )
3736fveq2d 5869 . . . . . . . . 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 3615 . . . . . . . . . . 11  |-  ( ( T `  y )  =  F  ->  (
( T `  y
)  \  _I  )  =  ( F  \  _I  ) )
4039dmeqd 5204 . . . . . . . . . 10  |-  ( ( T `  y )  =  F  ->  dom  ( ( T `  y )  \  _I  )  =  dom  ( F 
\  _I  ) )
41 pmtrfrn.p . . . . . . . . . 10  |-  P  =  dom  ( F  \  _I  )
4240, 41syl6eqr 2526 . . . . . . . . 9  |-  ( ( T `  y )  =  F  ->  dom  ( ( T `  y )  \  _I  )  =  P )
43 sseq1 3525 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( dom  ( ( T `
 y )  \  _I  )  C_  D  <->  P  C_  D
) )
44 breq1 4450 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( dom  ( ( T `
 y )  \  _I  )  ~~  2o  <->  P  ~~  2o ) )
4543, 443anbi23d 1302 . . . . . . . . . . 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 5865 . . . . . . . . . . . 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 2489 . . . . . . . . . 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 1195 . . . . . 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 2953 . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    \ cdif 3473    C_ wss 3476   (/)c0 3785   ifcif 3939   ~Pcpw 4010   {csn 4027   U.cuni 4245   class class class wbr 4447    |-> cmpt 4505    _I cid 4790   dom cdm 4999   ran crn 5000    Fn wfn 5582   ` cfv 5587   2oc2o 7124    ~~ cen 7513  pmTrspcpmtr 16269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-om 6680  df-1o 7130  df-2o 7131  df-er 7311  df-en 7517  df-fin 7520  df-pmtr 16270
This theorem is referenced by:  pmtrffv  16287  pmtrrn2  16288  pmtrfinv  16289  pmtrfmvdn0  16290  pmtrff1o  16291  pmtrfcnv  16292  pmtrfb  16293
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