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Theorem pmtrmvd 27501
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 27500 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
3 ffn 5405 . . 3  |-  ( ( T `  P ) : D --> D  -> 
( T `  P
)  Fn  D )
4 fndifnfp 26859 . . 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 27498 . . . . . 6  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  ( ( T `  P ) `  z
)  =  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) )
76neeq1d 2472 . . . . 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 3585 . . . . . . . 8  |-  ( -.  z  e.  P  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  z )
98necon1ai 2501 . . . . . . 7  |-  ( if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =/=  z  ->  z  e.  P )
10 iftrue 3584 . . . . . . . . . 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 6653 . . . . . . . . . . . 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 6496 . . . . . . . . . . . 12  |-  2o  =  suc  1o
1614, 15syl6breq 4078 . . . . . . . . . . 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 7107 . . . . . . . . . . 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 27483 . . . . . . . . . 10  |-  ( ( P  \  { z } )  ~~  1o  ->  U. ( P  \  { z } )  e.  ( P  \  { z } ) )
21 eldifsni 3763 . . . . . . . . . 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 2477 . . . . . . . 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 2792 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  { z  e.  D  |  ( ( T `  P
) `  z )  =/=  z }  =  {
z  e.  D  | 
z  e.  P }
)
29 incom 3374 . . . 4  |-  ( P  i^i  D )  =  ( D  i^i  P
)
30 dfin5 3173 . . . 4  |-  ( D  i^i  P )  =  { z  e.  D  |  z  e.  P }
3129, 30eqtri 2316 . . 3  |-  ( P  i^i  D )  =  { z  e.  D  |  z  e.  P }
3228, 31syl6eqr 2346 . 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 3179 . . 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 2332 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 1632    e. wcel 1696    =/= wne 2459   {crab 2560    \ cdif 3162    i^i cin 3164    C_ wss 3165   ifcif 3578   {csn 3653   U.cuni 3843   class class class wbr 4039    _I cid 4320   suc csuc 4410   omcom 4672   dom cdm 4705    Fn wfn 5266   -->wf 5267   ` cfv 5271   1oc1o 6488   2oc2o 6489    ~~ cen 6876  pmTrspcpmtr 27487
This theorem is referenced by:  pmtrfrn  27503  pmtrfb  27509  symggen  27514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-2o 6496  df-er 6676  df-en 6880  df-fin 6883  df-pmtr 27488
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