Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmtrfv Structured version   Unicode version

Theorem pmtrfv 27363
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 27362 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
32fveq1d 5722 . . 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 7106 . . . . . . 7  |-  Rel  ~~
87brrelexi 4910 . . . . . 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 4343 . . . . 5  |-  ( P  e.  _V  ->  ( P  \  { Z }
)  e.  _V )
11 uniexg 4698 . . . . 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 3790 . . . 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 2495 . . . . 5  |-  ( z  =  Z  ->  (
z  e.  P  <->  Z  e.  P ) )
16 sneq 3817 . . . . . . 7  |-  ( z  =  Z  ->  { z }  =  { Z } )
1716difeq2d 3457 . . . . . 6  |-  ( z  =  Z  ->  ( P  \  { z } )  =  ( P 
\  { Z }
) )
1817unieqd 4018 . . . . 5  |-  ( z  =  Z  ->  U. ( P  \  { z } )  =  U. ( P  \  { Z }
) )
19 id 20 . . . . 5  |-  ( z  =  Z  ->  z  =  Z )
2015, 18, 19ifbieq12d 3753 . . . 4  |-  ( z  =  Z  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  =  if ( Z  e.  P ,  U. ( P  \  { Z }
) ,  Z ) )
21 eqid 2435 . . . 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 5796 . . 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 2467 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 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    C_ wss 3312   ifcif 3731   {csn 3806   U.cuni 4007   class class class wbr 4204    e. cmpt 4258   ` cfv 5446   2oc2o 6710    ~~ cen 7098  pmTrspcpmtr 27352
This theorem is referenced by:  pmtrprfv  27364  pmtrmvd  27366  pmtrffv  27369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-en 7102  df-pmtr 27353
  Copyright terms: Public domain W3C validator