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Theorem pmtrprfv2 28611
Description: In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.)
Hypothesis
Ref Expression
pmtrprfv2.t  |-  T  =  (pmTrsp `  D )
Assertion
Ref Expression
pmtrprfv2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y
) )  ->  (
( T `  { X ,  Y }
) `  Y )  =  X )

Proof of Theorem pmtrprfv2
StepHypRef Expression
1 prcom 4050 . . . 4  |-  { Y ,  X }  =  { X ,  Y }
21fveq2i 5868 . . 3  |-  ( T `
 { Y ,  X } )  =  ( T `  { X ,  Y } )
32fveq1i 5866 . 2  |-  ( ( T `  { Y ,  X } ) `  Y )  =  ( ( T `  { X ,  Y }
) `  Y )
4 ancom 452 . . . . . . 7  |-  ( ( X  e.  D  /\  Y  e.  D )  <->  ( Y  e.  D  /\  X  e.  D )
)
5 necom 2677 . . . . . . 7  |-  ( X  =/=  Y  <->  Y  =/=  X )
64, 5anbi12i 703 . . . . . 6  |-  ( ( ( X  e.  D  /\  Y  e.  D
)  /\  X  =/=  Y )  <->  ( ( Y  e.  D  /\  X  e.  D )  /\  Y  =/=  X ) )
7 df-3an 987 . . . . . 6  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  <->  ( ( X  e.  D  /\  Y  e.  D )  /\  X  =/=  Y
) )
8 df-3an 987 . . . . . 6  |-  ( ( Y  e.  D  /\  X  e.  D  /\  Y  =/=  X )  <->  ( ( Y  e.  D  /\  X  e.  D )  /\  Y  =/=  X
) )
96, 7, 83bitr4i 281 . . . . 5  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  <->  ( Y  e.  D  /\  X  e.  D  /\  Y  =/= 
X ) )
109biimpi 198 . . . 4  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  -> 
( Y  e.  D  /\  X  e.  D  /\  Y  =/=  X
) )
1110anim2i 573 . . 3  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y
) )  ->  ( D  e.  V  /\  ( Y  e.  D  /\  X  e.  D  /\  Y  =/=  X
) ) )
12 pmtrprfv2.t . . . 4  |-  T  =  (pmTrsp `  D )
1312pmtrprfv 17094 . . 3  |-  ( ( D  e.  V  /\  ( Y  e.  D  /\  X  e.  D  /\  Y  =/=  X
) )  ->  (
( T `  { Y ,  X }
) `  Y )  =  X )
1411, 13syl 17 . 2  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y
) )  ->  (
( T `  { Y ,  X }
) `  Y )  =  X )
153, 14syl5eqr 2499 1  |-  ( ( D  e.  V  /\  ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y
) )  ->  (
( T `  { X ,  Y }
) `  Y )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   {cpr 3970   ` cfv 5582  pmTrspcpmtr 17082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-2o 7183  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pmtr 17083
This theorem is referenced by:  psgnfzto1stlem  28613
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