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Theorem pmtrfv 27395
Description: General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t  |-  T  =  (pmTrsp `  D )
Assertion
Ref Expression
pmtrfv  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  ( ( T `  P ) `  Z
)  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z
) )

Proof of Theorem pmtrfv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 pmtrfval.t . . . . 5  |-  T  =  (pmTrsp `  D )
21pmtrval 27394 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
32fveq1d 5527 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
( T `  P
) `  Z )  =  ( ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) ) `  Z ) )
43adantr 451 . 2  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  ( ( T `  P ) `  Z
)  =  ( ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) `
 Z ) )
5 simpr 447 . . 3  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  Z  e.  D )
6 simpl3 960 . . . . . 6  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  P  ~~  2o )
7 relen 6868 . . . . . . 7  |-  Rel  ~~
87brrelexi 4729 . . . . . 6  |-  ( P 
~~  2o  ->  P  e. 
_V )
96, 8syl 15 . . . . 5  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  P  e.  _V )
10 difexg 4162 . . . . 5  |-  ( P  e.  _V  ->  ( P  \  { Z }
)  e.  _V )
11 uniexg 4517 . . . . 5  |-  ( ( P  \  { Z } )  e.  _V  ->  U. ( P  \  { Z } )  e. 
_V )
129, 10, 113syl 18 . . . 4  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  U. ( P  \  { Z } )  e. 
_V )
13 ifexg 3624 . . . 4  |-  ( ( U. ( P  \  { Z } )  e. 
_V  /\  Z  e.  D )  ->  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z )  e.  _V )
1412, 5, 13syl2anc 642 . . 3  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  if ( Z  e.  P ,  U. ( P  \  { Z }
) ,  Z )  e.  _V )
15 eleq1 2343 . . . . 5  |-  ( z  =  Z  ->  (
z  e.  P  <->  Z  e.  P ) )
16 sneq 3651 . . . . . . 7  |-  ( z  =  Z  ->  { z }  =  { Z } )
1716difeq2d 3294 . . . . . 6  |-  ( z  =  Z  ->  ( P  \  { z } )  =  ( P 
\  { Z }
) )
1817unieqd 3838 . . . . 5  |-  ( z  =  Z  ->  U. ( P  \  { z } )  =  U. ( P  \  { Z }
) )
19 id 19 . . . . 5  |-  ( z  =  Z  ->  z  =  Z )
2015, 18, 19ifbieq12d 3587 . . . 4  |-  ( z  =  Z  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  if ( Z  e.  P ,  U. ( P  \  { Z }
) ,  Z ) )
21 eqid 2283 . . . 4  |-  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) )
2220, 21fvmptg 5600 . . 3  |-  ( ( Z  e.  D  /\  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z )  e.  _V )  ->  ( ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) ) `  Z )  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z ) )
235, 14, 22syl2anc 642 . 2  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  ( ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) `  Z
)  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z
) )
244, 23eqtrd 2315 1  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  ( ( T `  P ) `  Z
)  =  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ifcif 3565   {csn 3640   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   ` cfv 5255   2oc2o 6473    ~~ cen 6860  pmTrspcpmtr 27384
This theorem is referenced by:  pmtrprfv  27396  pmtrmvd  27398  pmtrffv  27401
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-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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-en 6864  df-pmtr 27385
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