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Theorem pmtrmvd 27376
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 27375 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
3 ffn 5592 . . 3  |-  ( ( T `  P ) : D --> D  -> 
( T `  P
)  Fn  D )
4 fndifnfp 26738 . . 3  |-  ( ( T `  P )  Fn  D  ->  dom  ( ( T `  P )  \  _I  )  =  { z  e.  D  |  (
( T `  P
) `  z )  =/=  z } )
52, 3, 43syl 19 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `  P )  \  _I  )  =  { z  e.  D  |  (
( T `  P
) `  z )  =/=  z } )
61pmtrfv 27373 . . . . . 6  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( T `  P ) `  z
)  =  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) )
76neeq1d 2615 . . . . 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 3747 . . . . . . . 8  |-  ( -.  z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  z )
98necon1ai 2647 . . . . . . 7  |-  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  ->  z  e.  P )
10 iftrue 3746 . . . . . . . . . 10  |-  ( z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  = 
U. ( P  \  { z } ) )
1110adantl 454 . . . . . . . . 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 6883 . . . . . . . . . . . 12  |-  1o  e.  om
1312a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  1o  e.  om )
14 simpl3 963 . . . . . . . . . . . 12  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  P  ~~  2o )
15 df-2o 6726 . . . . . . . . . . . 12  |-  2o  =  suc  1o
1614, 15syl6breq 4252 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  P  ~~  suc  1o )
17 simpr 449 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  z  e.  P )
18 dif1en 7342 . . . . . . . . . . 11  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
1913, 16, 17, 18syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
20 en1uniel 27358 . . . . . . . . . 10  |-  ( ( P  \  { z } )  ~~  1o  ->  U. ( P  \  { z } )  e.  ( P  \  { z } ) )
21 eldifsni 3929 . . . . . . . . . 10  |-  ( U. ( P  \  { z } )  e.  ( P  \  { z } )  ->  U. ( P  \  { z } )  =/=  z )
2219, 20, 213syl 19 . . . . . . . . 9  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  U. ( P  \  { z } )  =/=  z )
2311, 22eqnetrd 2620 . . . . . . . 8  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z )
2423ex 425 . . . . . . 7  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z ) )
259, 24impbid2 197 . . . . . 6  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  <->  z  e.  P
) )
2625adantr 453 . . . . 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 246 . . . 4  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( ( T `
 P ) `  z )  =/=  z  <->  z  e.  P ) )
2827rabbidva 2948 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  {
z  e.  D  | 
z  e.  P }
)
29 incom 3534 . . . 4  |-  ( P  i^i  D )  =  ( D  i^i  P
)
30 dfin5 3329 . . . 4  |-  ( D  i^i  P )  =  { z  e.  D  |  z  e.  P }
3129, 30eqtri 2457 . . 3  |-  ( P  i^i  D )  =  { z  e.  D  |  z  e.  P }
3228, 31syl6eqr 2487 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  ( P  i^i  D ) )
33 simp2 959 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  C_  D )
34 df-ss 3335 . . 3  |-  ( P 
C_  D  <->  ( P  i^i  D )  =  P )
3533, 34sylib 190 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( P  i^i  D )  =  P )
365, 32, 353eqtrd 2473 1  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `  P )  \  _I  )  =  P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   {crab 2710    \ cdif 3318    i^i cin 3320    C_ wss 3321   ifcif 3740   {csn 3815   U.cuni 4016   class class class wbr 4213    _I cid 4494   suc csuc 4584   omcom 4846   dom cdm 4879    Fn wfn 5450   -->wf 5451   ` cfv 5455   1oc1o 6718   2oc2o 6719    ~~ cen 7107  pmTrspcpmtr 27362
This theorem is referenced by:  pmtrfrn  27378  pmtrfb  27384  symggen  27389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-1o 6725  df-2o 6726  df-er 6906  df-en 7111  df-fin 7114  df-pmtr 27363
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