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Theorem pmtrfv 27066
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 27065 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
32fveq1d 5672 . . 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 452 . 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 448 . . 3  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  Z  e.  D )
6 simpl3 962 . . . . . 6  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  P  ~~  2o )
7 relen 7052 . . . . . . 7  |-  Rel  ~~
87brrelexi 4860 . . . . . 6  |-  ( P 
~~  2o  ->  P  e. 
_V )
96, 8syl 16 . . . . 5  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  P  e.  _V )
10 difexg 4294 . . . . 5  |-  ( P  e.  _V  ->  ( P  \  { Z }
)  e.  _V )
11 uniexg 4648 . . . . 5  |-  ( ( P  \  { Z } )  e.  _V  ->  U. ( P  \  { Z } )  e. 
_V )
129, 10, 113syl 19 . . . 4  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  U. ( P  \  { Z } )  e. 
_V )
13 ifexg 3743 . . . 4  |-  ( ( U. ( P  \  { Z } )  e. 
_V  /\  Z  e.  D )  ->  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z )  e.  _V )
1412, 5, 13syl2anc 643 . . 3  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  if ( Z  e.  P ,  U. ( P  \  { Z }
) ,  Z )  e.  _V )
15 eleq1 2449 . . . . 5  |-  ( z  =  Z  ->  (
z  e.  P  <->  Z  e.  P ) )
16 sneq 3770 . . . . . . 7  |-  ( z  =  Z  ->  { z }  =  { Z } )
1716difeq2d 3410 . . . . . 6  |-  ( z  =  Z  ->  ( P  \  { z } )  =  ( P 
\  { Z }
) )
1817unieqd 3970 . . . . 5  |-  ( z  =  Z  ->  U. ( P  \  { z } )  =  U. ( P  \  { Z }
) )
19 id 20 . . . . 5  |-  ( z  =  Z  ->  z  =  Z )
2015, 18, 19ifbieq12d 3706 . . . 4  |-  ( z  =  Z  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  if ( Z  e.  P ,  U. ( P  \  { Z }
) ,  Z ) )
21 eqid 2389 . . . 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 5745 . . 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 643 . 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 2421 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 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2901    \ cdif 3262    C_ wss 3265   ifcif 3684   {csn 3759   U.cuni 3959   class class class wbr 4155    e. cmpt 4209   ` cfv 5396   2oc2o 6656    ~~ cen 7044  pmTrspcpmtr 27055
This theorem is referenced by:  pmtrprfv  27067  pmtrmvd  27069  pmtrffv  27072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-en 7048  df-pmtr 27056
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