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Theorem pmtrmvd 27398
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 27397 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
3 ffn 5389 . . 3  |-  ( ( T `  P ) : D --> D  -> 
( T `  P
)  Fn  D )
4 fndifnfp 26756 . . 3  |-  ( ( T `  P )  Fn  D  ->  dom  ( ( T `  P )  \  _I  )  =  { z  e.  D  |  (
( T `  P
) `  z )  =/=  z } )
52, 3, 43syl 18 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  dom  ( ( T `  P )  \  _I  )  =  { z  e.  D  |  (
( T `  P
) `  z )  =/=  z } )
61pmtrfv 27395 . . . . . 6  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( T `  P ) `  z
)  =  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) )
76neeq1d 2459 . . . . 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 3572 . . . . . . . 8  |-  ( -.  z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  z )
98necon1ai 2488 . . . . . . 7  |-  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  ->  z  e.  P )
10 iftrue 3571 . . . . . . . . . 10  |-  ( z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  = 
U. ( P  \  { z } ) )
1110adantl 452 . . . . . . . . 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 6637 . . . . . . . . . . . 12  |-  1o  e.  om
1312a1i 10 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  1o  e.  om )
14 simpl3 960 . . . . . . . . . . . 12  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  P  ~~  2o )
15 df-2o 6480 . . . . . . . . . . . 12  |-  2o  =  suc  1o
1614, 15syl6breq 4062 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  P  ~~  suc  1o )
17 simpr 447 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  z  e.  P )
18 dif1en 7091 . . . . . . . . . . 11  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
1913, 16, 17, 18syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
20 en1uniel 27380 . . . . . . . . . 10  |-  ( ( P  \  { z } )  ~~  1o  ->  U. ( P  \  { z } )  e.  ( P  \  { z } ) )
21 eldifsni 3750 . . . . . . . . . 10  |-  ( U. ( P  \  { z } )  e.  ( P  \  { z } )  ->  U. ( P  \  { z } )  =/=  z )
2219, 20, 213syl 18 . . . . . . . . 9  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  U. ( P  \  { z } )  =/=  z )
2311, 22eqnetrd 2464 . . . . . . . 8  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z )
2423ex 423 . . . . . . 7  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z ) )
259, 24impbid2 195 . . . . . 6  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  <->  z  e.  P
) )
2625adantr 451 . . . . 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 244 . . . 4  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( ( T `
 P ) `  z )  =/=  z  <->  z  e.  P ) )
2827rabbidva 2779 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  {
z  e.  D  | 
z  e.  P }
)
29 incom 3361 . . . 4  |-  ( P  i^i  D )  =  ( D  i^i  P
)
30 dfin5 3160 . . . 4  |-  ( D  i^i  P )  =  { z  e.  D  |  z  e.  P }
3129, 30eqtri 2303 . . 3  |-  ( P  i^i  D )  =  { z  e.  D  |  z  e.  P }
3228, 31syl6eqr 2333 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  ( P  i^i  D ) )
33 simp2 956 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  C_  D )
34 df-ss 3166 . . 3  |-  ( P 
C_  D  <->  ( P  i^i  D )  =  P )
3533, 34sylib 188 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( P  i^i  D )  =  P )
365, 32, 353eqtrd 2319 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    \ cdif 3149    i^i cin 3151    C_ wss 3152   ifcif 3565   {csn 3640   U.cuni 3827   class class class wbr 4023    _I cid 4304   suc csuc 4394   omcom 4656   dom cdm 4689    Fn wfn 5250   -->wf 5251   ` cfv 5255   1oc1o 6472   2oc2o 6473    ~~ cen 6860  pmTrspcpmtr 27384
This theorem is referenced by:  pmtrfrn  27400  pmtrfb  27406  symggen  27411
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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