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Theorem pmtrf 17039
Description: Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t  |-  T  =  (pmTrsp `  D )
Assertion
Ref Expression
pmtrf  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )

Proof of Theorem pmtrf
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpll2 1045 . . . . 5  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  P  C_  D )
2 1onn 7295 . . . . . . . 8  |-  1o  e.  om
32a1i 11 . . . . . . 7  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  1o  e.  om )
4 simpll3 1046 . . . . . . . 8  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  P  ~~  2o )
5 df-2o 7138 . . . . . . . 8  |-  2o  =  suc  1o
64, 5syl6breq 4406 . . . . . . 7  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  P  ~~  suc  1o )
7 simpr 462 . . . . . . 7  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  z  e.  P )
8 dif1en 7757 . . . . . . 7  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
93, 6, 7, 8syl3anc 1264 . . . . . 6  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  ( P  \  { z } )  ~~  1o )
10 en1uniel 7595 . . . . . 6  |-  ( ( P  \  { z } )  ~~  1o  ->  U. ( P  \  { z } )  e.  ( P  \  { z } ) )
11 eldifi 3530 . . . . . 6  |-  ( U. ( P  \  { z } )  e.  ( P  \  { z } )  ->  U. ( P  \  { z } )  e.  P )
129, 10, 113syl 18 . . . . 5  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  U. ( P  \  { z } )  e.  P )
131, 12sseldd 3408 . . . 4  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  U. ( P  \  { z } )  e.  D )
14 simplr 760 . . . 4  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  -.  z  e.  P )  ->  z  e.  D )
1513, 14ifclda 3886 . . 3  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  e.  D )
16 eqid 2428 . . 3  |-  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) )
1715, 16fmptd 6005 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) : D --> D )
18 pmtrfval.t . . . 4  |-  T  =  (pmTrsp `  D )
1918pmtrval 17035 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
2019feq1d 5675 . 2  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  (
( T `  P
) : D --> D  <->  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  {
z } ) ,  z ) ) : D --> D ) )
2117, 20mpbird 235 1  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P ) : D --> D )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    \ cdif 3376    C_ wss 3379   ifcif 3854   {csn 3941   U.cuni 4162   class class class wbr 4366    |-> cmpt 4425   suc csuc 5387   -->wf 5540   ` cfv 5544   omcom 6650   1oc1o 7130   2oc2o 7131    ~~ cen 7521  pmTrspcpmtr 17025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-om 6651  df-1o 7137  df-2o 7138  df-er 7318  df-en 7525  df-fin 7528  df-pmtr 17026
This theorem is referenced by:  pmtrmvd  17040  pmtrfinv  17045  pmtrff1o  17047  pmtrfcnv  17048  pmtr3ncomlem1  17057  mdetralt  19575  mdetunilem7  19585
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