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Theorem pmtrval 26542
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 26541 . . . 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 5565 . . 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 4211 . . . . . 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 4063 . . . . 5  |-  ( y  =  P  ->  (
y  ~~  2o  <->  P  ~~  2o ) )
109elrab 2957 . . . 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 5786 . . . 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 2377 . . . . . 6  |-  ( p  =  P  ->  (
z  e.  p  <->  z  e.  P ) )
15 difeq1 3321 . . . . . . 7  |-  ( p  =  P  ->  (
p  \  { z } )  =  ( P  \  { z } ) )
1615unieqd 3875 . . . . . 6  |-  ( p  =  P  ->  U. (
p  \  { z } )  =  U. ( P  \  { z } ) )
17 eqidd 2317 . . . . . 6  |-  ( p  =  P  ->  z  =  z )
1814, 16, 17ifbieq12d 3621 . . . . 5  |-  ( p  =  P  ->  if ( z  e.  p ,  U. ( p  \  { z } ) ,  z )  =  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) )
1918mpteq2dv 4144 . . . 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 2316 . . . 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 5638 . . 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 2348 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 1633    e. wcel 1701   {crab 2581   _Vcvv 2822    \ cdif 3183    C_ wss 3186   ifcif 3599   ~Pcpw 3659   {csn 3674   U.cuni 3864   class class class wbr 4060    e. cmpt 4114   ` cfv 5292   2oc2o 6515    ~~ cen 6903  pmTrspcpmtr 26532
This theorem is referenced by:  pmtrfv  26543  pmtrf  26545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-pmtr 26533
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