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Theorem pmtrprfv2 28685
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 4041 . . . 4  |-  { Y ,  X }  =  { X ,  Y }
21fveq2i 5882 . . 3  |-  ( T `
 { Y ,  X } )  =  ( T `  { X ,  Y } )
32fveq1i 5880 . 2  |-  ( ( T `  { Y ,  X } ) `  Y )  =  ( ( T `  { X ,  Y }
) `  Y )
4 ancom 457 . . . . . . 7  |-  ( ( X  e.  D  /\  Y  e.  D )  <->  ( Y  e.  D  /\  X  e.  D )
)
5 necom 2696 . . . . . . 7  |-  ( X  =/=  Y  <->  Y  =/=  X )
64, 5anbi12i 711 . . . . . 6  |-  ( ( ( X  e.  D  /\  Y  e.  D
)  /\  X  =/=  Y )  <->  ( ( Y  e.  D  /\  X  e.  D )  /\  Y  =/=  X ) )
7 df-3an 1009 . . . . . 6  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  <->  ( ( X  e.  D  /\  Y  e.  D )  /\  X  =/=  Y
) )
8 df-3an 1009 . . . . . 6  |-  ( ( Y  e.  D  /\  X  e.  D  /\  Y  =/=  X )  <->  ( ( Y  e.  D  /\  X  e.  D )  /\  Y  =/=  X
) )
96, 7, 83bitr4i 285 . . . . 5  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  <->  ( Y  e.  D  /\  X  e.  D  /\  Y  =/= 
X ) )
109biimpi 199 . . . 4  |-  ( ( X  e.  D  /\  Y  e.  D  /\  X  =/=  Y )  -> 
( Y  e.  D  /\  X  e.  D  /\  Y  =/=  X
) )
1110anim2i 579 . . 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 17172 . . 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 2519 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   {cpr 3961   ` cfv 5589  pmTrspcpmtr 17160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-2o 7201  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pmtr 17161
This theorem is referenced by:  psgnfzto1stlem  28687
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