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Theorem pmtrfv 16676
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 16675 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
32fveq1d 5850 . . 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 463 . 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 459 . . 3  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  Z  e.  D )
6 simpl3 999 . . . . 5  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  P  ~~  2o )
7 relen 7514 . . . . . 6  |-  Rel  ~~
87brrelexi 5029 . . . . 5  |-  ( P 
~~  2o  ->  P  e. 
_V )
9 difexg 4585 . . . . 5  |-  ( P  e.  _V  ->  ( P  \  { Z }
)  e.  _V )
10 uniexg 6570 . . . . 5  |-  ( ( P  \  { Z } )  e.  _V  ->  U. ( P  \  { Z } )  e. 
_V )
116, 8, 9, 104syl 21 . . . 4  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  U. ( P  \  { Z } )  e. 
_V )
12 ifexg 3998 . . . 4  |-  ( ( U. ( P  \  { Z } )  e. 
_V  /\  Z  e.  D )  ->  if ( Z  e.  P ,  U. ( P  \  { Z } ) ,  Z )  e.  _V )
1311, 5, 12syl2anc 659 . . 3  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  Z  e.  D )  ->  if ( Z  e.  P ,  U. ( P  \  { Z }
) ,  Z )  e.  _V )
14 eleq1 2526 . . . . 5  |-  ( z  =  Z  ->  (
z  e.  P  <->  Z  e.  P ) )
15 sneq 4026 . . . . . . 7  |-  ( z  =  Z  ->  { z }  =  { Z } )
1615difeq2d 3608 . . . . . 6  |-  ( z  =  Z  ->  ( P  \  { z } )  =  ( P 
\  { Z }
) )
1716unieqd 4245 . . . . 5  |-  ( z  =  Z  ->  U. ( P  \  { z } )  =  U. ( P  \  { Z }
) )
18 id 22 . . . . 5  |-  ( z  =  Z  ->  z  =  Z )
1914, 17, 18ifbieq12d 3956 . . . 4  |-  ( z  =  Z  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  if ( Z  e.  P ,  U. ( P  \  { Z }
) ,  Z ) )
20 eqid 2454 . . . 4  |-  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) )
2119, 20fvmptg 5929 . . 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 ) )
225, 13, 21syl2anc 659 . 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
) )
234, 22eqtrd 2495 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    \ cdif 3458    C_ wss 3461   ifcif 3929   {csn 4016   U.cuni 4235   class class class wbr 4439    |-> cmpt 4497   ` cfv 5570   2oc2o 7116    ~~ cen 7506  pmTrspcpmtr 16665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-en 7510  df-pmtr 16666
This theorem is referenced by:  pmtrprfv  16677  pmtrprfv3  16678  pmtrmvd  16680  pmtrffv  16683
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