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Theorem pmtrprfv3 16681
Description: In a transposition of two given points, all other points are mapped to themselves. (Contributed by AV, 17-Mar-2019.)
Hypothesis
Ref Expression
pmtrfval.t  |-  T  =  (pmTrsp `  D )
Assertion
Ref Expression
pmtrprfv3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { X ,  Y }
) `  Z )  =  Z )

Proof of Theorem pmtrprfv3
StepHypRef Expression
1 simp1 994 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  D  e.  V )
2 simp1 994 . . . . 5  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  X  e.  D )
323ad2ant2 1016 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  X  e.  D )
4 simp22 1028 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Y  e.  D )
5 prssi 4172 . . . 4  |-  ( ( X  e.  D  /\  Y  e.  D )  ->  { X ,  Y }  C_  D )
63, 4, 5syl2anc 659 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { X ,  Y }  C_  D
)
7 pr2nelem 8373 . . . . . . . . 9  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  ->  { X ,  Y }  ~~  2o )
873expia 1196 . . . . . . . 8  |-  ( ( X  e.  D  /\  Y  e.  D )  ->  ( X  =/=  Y  ->  { X ,  Y }  ~~  2o ) )
983adant3 1014 . . . . . . 7  |-  ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D )  ->  ( X  =/=  Y  ->  { X ,  Y }  ~~  2o ) )
109com12 31 . . . . . 6  |-  ( X  =/=  Y  ->  (
( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  ->  { X ,  Y }  ~~  2o ) )
11103ad2ant1 1015 . . . . 5  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  (
( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  ->  { X ,  Y }  ~~  2o ) )
1211impcom 428 . . . 4  |-  ( ( ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { X ,  Y }  ~~  2o )
13123adant1 1012 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  { X ,  Y }  ~~  2o )
14 simp23 1029 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Z  e.  D )
15 pmtrfval.t . . . 4  |-  T  =  (pmTrsp `  D )
1615pmtrfv 16679 . . 3  |-  ( ( ( D  e.  V  /\  { X ,  Y }  C_  D  /\  { X ,  Y }  ~~  2o )  /\  Z  e.  D )  ->  (
( T `  { X ,  Y }
) `  Z )  =  if ( Z  e. 
{ X ,  Y } ,  U. ( { X ,  Y }  \  { Z } ) ,  Z ) )
171, 6, 13, 14, 16syl31anc 1229 . 2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { X ,  Y }
) `  Z )  =  if ( Z  e. 
{ X ,  Y } ,  U. ( { X ,  Y }  \  { Z } ) ,  Z ) )
18 necom 2723 . . . . . . 7  |-  ( X  =/=  Z  <->  Z  =/=  X )
1918biimpi 194 . . . . . 6  |-  ( X  =/=  Z  ->  Z  =/=  X )
20193ad2ant2 1016 . . . . 5  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  Z  =/=  X )
21203ad2ant3 1017 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Z  =/=  X )
22 necom 2723 . . . . . . 7  |-  ( Y  =/=  Z  <->  Z  =/=  Y )
2322biimpi 194 . . . . . 6  |-  ( Y  =/=  Z  ->  Z  =/=  Y )
24233ad2ant3 1017 . . . . 5  |-  ( ( X  =/=  Y  /\  X  =/=  Z  /\  Y  =/=  Z )  ->  Z  =/=  Y )
25243ad2ant3 1017 . . . 4  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  Z  =/=  Y )
2621, 25nelprd 4038 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  -.  Z  e.  { X ,  Y } )
2726iffalsed 3940 . 2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  if ( Z  e.  { X ,  Y } ,  U. ( { X ,  Y }  \  { Z }
) ,  Z )  =  Z )
2817, 27eqtrd 2495 1  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  Z  e.  D
)  /\  ( X  =/=  Y  /\  X  =/= 
Z  /\  Y  =/=  Z ) )  ->  (
( T `  { X ,  Y }
) `  Z )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458    C_ wss 3461   ifcif 3929   {csn 4016   {cpr 4018   U.cuni 4235   class class class wbr 4439   ` cfv 5570   2oc2o 7116    ~~ cen 7506  pmTrspcpmtr 16668
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-dom 7511  df-sdom 7512  df-pmtr 16669
This theorem is referenced by:  pmtr3ncomlem1  16700
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