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Theorem pmtrf 16679
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 1034 . . . . 5  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  P  C_  D )
2 1onn 7280 . . . . . . . 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 1035 . . . . . . . 8  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  P  ~~  2o )
5 df-2o 7123 . . . . . . . 8  |-  2o  =  suc  1o
64, 5syl6breq 4478 . . . . . . 7  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  P  ~~  suc  1o )
7 simpr 459 . . . . . . 7  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  z  e.  P )
8 dif1en 7745 . . . . . . 7  |-  ( ( 1o  e.  om  /\  P  ~~  suc  1o  /\  z  e.  P )  ->  ( P  \  {
z } )  ~~  1o )
93, 6, 7, 8syl3anc 1226 . . . . . 6  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  ( P  \  { z } )  ~~  1o )
10 en1uniel 7580 . . . . . 6  |-  ( ( P  \  { z } )  ~~  1o  ->  U. ( P  \  { z } )  e.  ( P  \  { z } ) )
11 eldifi 3612 . . . . . 6  |-  ( U. ( P  \  { z } )  e.  ( P  \  { z } )  ->  U. ( P  \  { z } )  e.  P )
129, 10, 113syl 20 . . . . 5  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  U. ( P  \  { z } )  e.  P )
131, 12sseldd 3490 . . . 4  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  z  e.  P )  ->  U. ( P  \  { z } )  e.  D )
14 simplr 753 . . . 4  |-  ( ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D
)  /\  -.  z  e.  P )  ->  z  e.  D )
1513, 14ifclda 3961 . . 3  |-  ( ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  /\  z  e.  D )  ->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z )  e.  D )
16 eqid 2454 . . 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 6031 . 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 16675 . . 3  |-  ( ( D  e.  V  /\  P  C_  D  /\  P  ~~  2o )  ->  ( T `  P )  =  ( z  e.  D  |->  if ( z  e.  P ,  U. ( P  \  { z } ) ,  z ) ) )
2019feq1d 5699 . 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 232 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    \ cdif 3458    C_ wss 3461   ifcif 3929   {csn 4016   U.cuni 4235   class class class wbr 4439    |-> cmpt 4497   suc csuc 4869   -->wf 5566   ` cfv 5570   omcom 6673   1oc1o 7115   2oc2o 7116    ~~ cen 7506  pmTrspcpmtr 16665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-om 6674  df-1o 7122  df-2o 7123  df-er 7303  df-en 7510  df-fin 7513  df-pmtr 16666
This theorem is referenced by:  pmtrmvd  16680  pmtrfinv  16685  pmtrff1o  16687  pmtrfcnv  16688  pmtr3ncomlem1  16697  mdetralt  19277  mdetunilem7  19287
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