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Theorem pmtrfrn 27368
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 3624 . . . 4  |-  -.  F  e.  (/)
2 pmtrrn.r . . . . . 6  |-  R  =  ran  T
3 pmtrrn.t . . . . . . . . 9  |-  T  =  (pmTrsp `  D )
4 fvprc 5714 . . . . . . . . 9  |-  ( -.  D  e.  _V  ->  (pmTrsp `  D )  =  (/) )
53, 4syl5eq 2479 . . . . . . . 8  |-  ( -.  D  e.  _V  ->  T  =  (/) )
65rneqd 5089 . . . . . . 7  |-  ( -.  D  e.  _V  ->  ran 
T  =  ran  (/) )
7 rn0 5119 . . . . . . 7  |-  ran  (/)  =  (/)
86, 7syl6eq 2483 . . . . . 6  |-  ( -.  D  e.  _V  ->  ran 
T  =  (/) )
92, 8syl5eq 2479 . . . . 5  |-  ( -.  D  e.  _V  ->  R  =  (/) )
109eleq2d 2502 . . . 4  |-  ( -.  D  e.  _V  ->  ( F  e.  R  <->  F  e.  (/) ) )
111, 10mtbiri 295 . . 3  |-  ( -.  D  e.  _V  ->  -.  F  e.  R )
1211con4i 124 . 2  |-  ( F  e.  R  ->  D  e.  _V )
13 mptexg 5957 . . . . . . . 8  |-  ( D  e.  _V  ->  (
z  e.  D  |->  if ( z  e.  w ,  U. ( w  \  { z } ) ,  z ) )  e.  _V )
1413ralrimivw 2782 . . . . . . 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 2435 . . . . . . . 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 5563 . . . . . . 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 27361 . . . . . . 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 5528 . . . . . 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 224 . . . . 5  |-  ( D  e.  _V  ->  T  Fn  { x  e.  ~P D  |  x  ~~  2o } )
21 fvelrnb 5766 . . . . 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 2499 . . . 4  |-  ( F  e.  R  <->  F  e.  ran  T )
24 breq1 4207 . . . . . 6  |-  ( x  =  y  ->  (
x  ~~  2o  <->  y  ~~  2o ) )
2524rexrab 3090 . . . . 5  |-  ( E. y  e.  { x  e.  ~P D  |  x 
~~  2o }  ( T `  y )  =  F  <->  E. y  e.  ~P  D ( y  ~~  2o  /\  ( T `  y )  =  F ) )
2625bicomi 194 . . . 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 280 . . 3  |-  ( D  e.  _V  ->  ( F  e.  R  <->  E. y  e.  ~P  D ( y 
~~  2o  /\  ( T `  y )  =  F ) ) )
28 elpwi 3799 . . . . 5  |-  ( y  e.  ~P D  -> 
y  C_  D )
29 simp1 957 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  D  e.  _V )
303pmtrmvd 27366 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  =  y )
31 simp2 958 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  C_  D )
3230, 31eqsstrd 3374 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  C_  D )
33 simp3 959 . . . . . . . . . . 11  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  ~~  2o )
3430, 33eqbrtrd 4224 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  dom  ( ( T `  y )  \  _I  )  ~~  2o )
3529, 32, 343jca 1134 . . . . . . . . 9  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  ( D  e.  _V  /\  dom  ( ( T `  y )  \  _I  )  C_  D  /\  dom  ( ( T `  y )  \  _I  )  ~~  2o ) )
3630eqcomd 2440 . . . . . . . . . 10  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  y  =  dom  ( ( T `
 y )  \  _I  ) )
3736fveq2d 5724 . . . . . . . . 9  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  ( T `  y )  =  ( T `  dom  ( ( T `  y )  \  _I  ) ) )
3835, 37jca 519 . . . . . . . 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 3450 . . . . . . . . . . 11  |-  ( ( T `  y )  =  F  ->  (
( T `  y
)  \  _I  )  =  ( F  \  _I  ) )
4039dmeqd 5064 . . . . . . . . . 10  |-  ( ( T `  y )  =  F  ->  dom  ( ( T `  y )  \  _I  )  =  dom  ( F 
\  _I  ) )
41 pmtrfrn.p . . . . . . . . . 10  |-  P  =  dom  ( F  \  _I  )
4240, 41syl6eqr 2485 . . . . . . . . 9  |-  ( ( T `  y )  =  F  ->  dom  ( ( T `  y )  \  _I  )  =  P )
43 sseq1 3361 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( dom  ( ( T `
 y )  \  _I  )  C_  D  <->  P  C_  D
) )
44 breq1 4207 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( dom  ( ( T `
 y )  \  _I  )  ~~  2o  <->  P  ~~  2o ) )
4543, 443anbi23d 1257 . . . . . . . . . . 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 453 . . . . . . . . . 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 444 . . . . . . . . . . 11  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( T `  y
)  =  F )
48 fveq2 5720 . . . . . . . . . . . 12  |-  ( dom  ( ( T `  y )  \  _I  )  =  P  ->  ( T `  dom  (
( T `  y
)  \  _I  )
)  =  ( T `
 P ) )
4948adantl 453 . . . . . . . . . . 11  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( T `  dom  ( ( T `  y )  \  _I  ) )  =  ( T `  P ) )
5047, 49eqeq12d 2449 . . . . . . . . . 10  |-  ( ( ( T `  y
)  =  F  /\  dom  ( ( T `  y )  \  _I  )  =  P )  ->  ( ( T `  y )  =  ( T `  dom  (
( T `  y
)  \  _I  )
)  <->  F  =  ( T `  P )
) )
5146, 50anbi12d 692 . . . . . . . . 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 650 . . . . . . . 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 212 . . . . . . 7  |-  ( ( D  e.  _V  /\  y  C_  D  /\  y  ~~  2o )  ->  (
( T `  y
)  =  F  -> 
( ( D  e. 
_V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) )
54533exp 1152 . . . . . 6  |-  ( D  e.  _V  ->  (
y  C_  D  ->  ( y  ~~  2o  ->  ( ( T `  y
)  =  F  -> 
( ( D  e. 
_V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) ) ) )
5554imp4a 573 . . . . 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 30 . . . 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 2821 . . 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 207 . 2  |-  ( D  e.  _V  ->  ( F  e.  R  ->  ( ( D  e.  _V  /\  P  C_  D  /\  P  ~~  2o )  /\  F  =  ( T `  P ) ) ) )
5912, 58mpcom 34 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309    C_ wss 3312   (/)c0 3620   ifcif 3731   ~Pcpw 3791   {csn 3806   U.cuni 4007   class class class wbr 4204    e. cmpt 4258    _I cid 4485   dom cdm 4870   ran crn 4871    Fn wfn 5441   ` cfv 5446   2oc2o 6710    ~~ cen 7098  pmTrspcpmtr 27352
This theorem is referenced by:  pmtrffv  27369  pmtrfinv  27370  pmtrfmvdn0  27371  pmtrff1o  27372  pmtrfcnv  27373  pmtrfb  27374
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1o 6716  df-2o 6717  df-er 6897  df-en 7102  df-fin 7105  df-pmtr 27353
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