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Theorem pmtrval 27394
Description: A generated transposition, expressed in a symmetric form. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t  |-  T  =  (pmTrsp `  D )
Assertion
Ref Expression
pmtrval  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
Distinct variable groups:    z, D    z, T    z, P    z, V

Proof of Theorem pmtrval
Dummy variables  p  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pmtrfval.t . . . . 5  |-  T  =  (pmTrsp `  D )
21pmtrfval 27393 . . . 4  |-  ( D  e.  V  ->  T  =  ( p  e. 
{ y  e.  ~P D  |  y  ~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z ) ) ) )
32fveq1d 5527 . . 3  |-  ( D  e.  V  ->  ( T `  P )  =  ( ( p  e.  { y  e. 
~P D  |  y 
~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z ) ) ) `  P ) )
433ad2ant1 976 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( ( p  e.  { y  e. 
~P D  |  y 
~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z ) ) ) `  P ) )
5 elpw2g 4174 . . . . . 6  |-  ( D  e.  V  ->  ( P  e.  ~P D  <->  P 
C_  D ) )
65biimpar 471 . . . . 5  |-  ( ( D  e.  V  /\  P  C_  D )  ->  P  e.  ~P D
)
763adant3 975 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  ~P D )
8 simp3 957 . . . 4  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  ~~  2o )
9 breq1 4026 . . . . 5  |-  ( y  =  P  ->  (
y  ~~  2o  <->  P  ~~  2o ) )
109elrab 2923 . . . 4  |-  ( P  e.  { y  e. 
~P D  |  y 
~~  2o }  <->  ( P  e.  ~P D  /\  P  ~~  2o ) )
117, 8, 10sylanbrc 645 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  P  e.  { y  e.  ~P D  |  y  ~~  2o } )
12 mptexg 5745 . . . 4  |-  ( D  e.  V  ->  (
z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) )  e.  _V )
13123ad2ant1 976 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) )  e.  _V )
14 eleq2 2344 . . . . . 6  |-  ( p  =  P  ->  (
z  e.  p  <->  z  e.  P ) )
15 difeq1 3287 . . . . . . 7  |-  ( p  =  P  ->  (
p  \  { z } )  =  ( P  \  { z } ) )
1615unieqd 3838 . . . . . 6  |-  ( p  =  P  ->  U. (
p  \  { z } )  =  U. ( P  \  { z } ) )
17 eqidd 2284 . . . . . 6  |-  ( p  =  P  ->  z  =  z )
1814, 16, 17ifbieq12d 3587 . . . . 5  |-  ( p  =  P  ->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z )  =  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) )
1918mpteq2dv 4107 . . . 4  |-  ( p  =  P  ->  (
z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z ) )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
20 eqid 2283 . . . 4  |-  ( p  e.  { y  e. 
~P D  |  y 
~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z ) ) )  =  ( p  e.  { y  e. 
~P D  |  y 
~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z ) ) )
2119, 20fvmptg 5600 . . 3  |-  ( ( P  e.  { y  e.  ~P D  | 
y  ~~  2o }  /\  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) )  e.  _V )  ->  ( ( p  e. 
{ y  e.  ~P D  |  y  ~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z ) ) ) `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
2211, 13, 21syl2anc 642 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
( p  e.  {
y  e.  ~P D  |  y  ~~  2o }  |->  ( z  e.  D  |->  if ( z  e.  p ,  U. (
p  \  { z } ) ,  z ) ) ) `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
234, 22eqtrd 2315 1  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ifcif 3565   ~Pcpw 3625   {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:  pmtrfv  27395  pmtrf  27397
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-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
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-pmtr 27385
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