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Theorem pmtrmvd 16274
Description: A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t  |-  T  =  (pmTrsp `  D )
Assertion
Ref Expression
pmtrmvd  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `  P )  \  _I  )  =  P )

Proof of Theorem pmtrmvd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 pmtrfval.t . . . 4  |-  T  =  (pmTrsp `  D )
21pmtrf 16273 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
3 ffn 5729 . . 3  |-  ( ( T `  P ) : D --> D  -> 
( T `  P
)  Fn  D )
4 fndifnfp 6088 . . 3  |-  ( ( T `  P )  Fn  D  ->  dom  ( ( T `  P )  \  _I  )  =  { z  e.  D  |  (
( T `  P
) `  z )  =/=  z } )
52, 3, 43syl 20 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `  P )  \  _I  )  =  { z  e.  D  |  (
( T `  P
) `  z )  =/=  z } )
61pmtrfv 16270 . . . . . 6  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( T `  P ) `  z
)  =  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) )
76neeq1d 2744 . . . . 5  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( ( T `
 P ) `  z )  =/=  z  <->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z ) )
8 iffalse 3948 . . . . . . . 8  |-  ( -.  z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  z )
98necon1ai 2698 . . . . . . 7  |-  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  ->  z  e.  P )
10 iftrue 3945 . . . . . . . . . 10  |-  ( z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  = 
U. ( P  \  { z } ) )
1110adantl 466 . . . . . . . . 9  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  U. ( P 
\  { z } ) )
12 1onn 7285 . . . . . . . . . . . 12  |-  1o  e.  om
1312a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  1o  e.  om )
14 simpl3 1001 . . . . . . . . . . . 12  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  P  ~~  2o )
15 df-2o 7128 . . . . . . . . . . . 12  |-  2o  =  suc  1o
1614, 15syl6breq 4486 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  P  ~~  suc  1o )
17 simpr 461 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  z  e.  P )
18 dif1en 7749 . . . . . . . . . . 11  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
1913, 16, 17, 18syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
20 en1uniel 7584 . . . . . . . . . 10  |-  ( ( P  \  { z } )  ~~  1o  ->  U. ( P  \  { z } )  e.  ( P  \  { z } ) )
21 eldifsni 4153 . . . . . . . . . 10  |-  ( U. ( P  \  { z } )  e.  ( P  \  { z } )  ->  U. ( P  \  { z } )  =/=  z )
2219, 20, 213syl 20 . . . . . . . . 9  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  U. ( P  \  { z } )  =/=  z )
2311, 22eqnetrd 2760 . . . . . . . 8  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z )
2423ex 434 . . . . . . 7  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z ) )
259, 24impbid2 204 . . . . . 6  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  <->  z  e.  P
) )
2625adantr 465 . . . . 5  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  <->  z  e.  P ) )
277, 26bitrd 253 . . . 4  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( ( T `
 P ) `  z )  =/=  z  <->  z  e.  P ) )
2827rabbidva 3104 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  {
z  e.  D  | 
z  e.  P }
)
29 incom 3691 . . . 4  |-  ( P  i^i  D )  =  ( D  i^i  P
)
30 dfin5 3484 . . . 4  |-  ( D  i^i  P )  =  { z  e.  D  |  z  e.  P }
3129, 30eqtri 2496 . . 3  |-  ( P  i^i  D )  =  { z  e.  D  |  z  e.  P }
3228, 31syl6eqr 2526 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  ( P  i^i  D ) )
33 simp2 997 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  C_  D )
34 df-ss 3490 . . 3  |-  ( P 
C_  D  <->  ( P  i^i  D )  =  P )
3533, 34sylib 196 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( P  i^i  D )  =  P )
365, 32, 353eqtrd 2512 1  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `  P )  \  _I  )  =  P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818    \ cdif 3473    i^i cin 3475    C_ wss 3476   ifcif 3939   {csn 4027   U.cuni 4245   class class class wbr 4447    _I cid 4790   suc csuc 4880   dom cdm 4999    Fn wfn 5581   -->wf 5582   ` cfv 5586   omcom 6678   1oc1o 7120   2oc2o 7121    ~~ cen 7510  pmTrspcpmtr 16259
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 6574
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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-1o 7127  df-2o 7128  df-er 7308  df-en 7514  df-fin 7517  df-pmtr 16260
This theorem is referenced by:  pmtrfrn  16276  pmtrfb  16283  symggen  16288  pmtrdifellem2  16295  mdetralt  18874  mdetunilem7  18884
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