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Theorem pmtrfv 27498
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 27497 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
32fveq1d 5543 . . 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 6884 . . . . . . 7  |-  Rel  ~~
87brrelexi 4745 . . . . . 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 4178 . . . . 5  |-  ( P  e.  _V  ->  ( P  \  { Z }
)  e.  _V )
11 uniexg 4533 . . . . 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 3637 . . . 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 2356 . . . . 5  |-  ( z  =  Z  ->  (
z  e.  P  <->  Z  e.  P ) )
16 sneq 3664 . . . . . . 7  |-  ( z  =  Z  ->  { z }  =  { Z } )
1716difeq2d 3307 . . . . . 6  |-  ( z  =  Z  ->  ( P  \  { z } )  =  ( P 
\  { Z }
) )
1817unieqd 3854 . . . . 5  |-  ( z  =  Z  ->  U. ( P  \  { z } )  =  U. ( P  \  { Z }
) )
19 id 19 . . . . 5  |-  ( z  =  Z  ->  z  =  Z )
2015, 18, 19ifbieq12d 3600 . . . 4  |-  ( z  =  Z  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  if ( Z  e.  P ,  U. ( P  \  { Z }
) ,  Z ) )
21 eqid 2296 . . . 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 5616 . . 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 2328 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 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   ifcif 3578   {csn 3653   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   ` cfv 5271   2oc2o 6489    ~~ cen 6876  pmTrspcpmtr 27487
This theorem is referenced by:  pmtrprfv  27499  pmtrmvd  27501  pmtrffv  27504
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-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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-en 6880  df-pmtr 27488
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