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Theorem pmtrmvd 15960
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 15959 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
3 ffn 5557 . . 3  |-  ( ( T `  P ) : D --> D  -> 
( T `  P
)  Fn  D )
4 fndifnfp 5905 . . 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 15956 . . . . . 6  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( T `  P ) `  z
)  =  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) )
76neeq1d 2619 . . . . 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 3797 . . . . . . . 8  |-  ( -.  z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  z )
98necon1ai 2651 . . . . . . 7  |-  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  ->  z  e.  P )
10 iftrue 3795 . . . . . . . . . 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 7076 . . . . . . . . . . . 12  |-  1o  e.  om
1312a1i 11 . . . . . . . . . . 11  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  1o  e.  om )
14 simpl3 993 . . . . . . . . . . . 12  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  P  ~~  2o )
15 df-2o 6919 . . . . . . . . . . . 12  |-  2o  =  suc  1o
1614, 15syl6breq 4329 . . . . . . . . . . 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 7543 . . . . . . . . . . 11  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
1913, 16, 17, 18syl3anc 1218 . . . . . . . . . 10  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
20 en1uniel 7379 . . . . . . . . . 10  |-  ( ( P  \  { z } )  ~~  1o  ->  U. ( P  \  { z } )  e.  ( P  \  { z } ) )
21 eldifsni 3999 . . . . . . . . . 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 2624 . . . . . . . 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 2961 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  {
z  e.  D  | 
z  e.  P }
)
29 incom 3541 . . . 4  |-  ( P  i^i  D )  =  ( D  i^i  P
)
30 dfin5 3334 . . . 4  |-  ( D  i^i  P )  =  { z  e.  D  |  z  e.  P }
3129, 30eqtri 2461 . . 3  |-  ( P  i^i  D )  =  { z  e.  D  |  z  e.  P }
3228, 31syl6eqr 2491 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  ( P  i^i  D ) )
33 simp2 989 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  C_  D )
34 df-ss 3340 . . 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 2477 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 965    = wceq 1369    e. wcel 1756    =/= wne 2604   {crab 2717    \ cdif 3323    i^i cin 3325    C_ wss 3326   ifcif 3789   {csn 3875   U.cuni 4089   class class class wbr 4290    _I cid 4629   suc csuc 4719   dom cdm 4838    Fn wfn 5411   -->wf 5412   ` cfv 5416   omcom 6474   1oc1o 6911   2oc2o 6912    ~~ cen 7305  pmTrspcpmtr 15945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-om 6475  df-1o 6918  df-2o 6919  df-er 7099  df-en 7309  df-fin 7312  df-pmtr 15946
This theorem is referenced by:  pmtrfrn  15962  pmtrfb  15969  symggen  15974  pmtrdifellem2  15981  mdetralt  18412  mdetunilem7  18422
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