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

Step	Hyp	Ref	Expression
1		prcom 4041	. . . 4 |- { Y , X } = { X , Y }
2	1	fveq2i 5882	. . 3 |- ( T ` { Y , X } ) = ( T ` { X , Y } )
3	2	fveq1i 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 )
6	4, 5	anbi12i 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 ) )
9	6, 7, 8	3bitr4i 285	. . . . 5 |- ( ( X e. D /\ Y e. D /\ X =/= Y ) <-> ( Y e. D /\ X e. D /\ Y =/= X ) )
10	9	biimpi 199	. . . 4 |- ( ( X e. D /\ Y e. D /\ X =/= Y ) -> ( Y e. D /\ X e. D /\ Y =/= X ) )
11	10	anim2i 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 )
13	12	pmtrprfv 17172	. . 3 |- ( ( D e. V /\ ( Y e. D /\ X e. D /\ Y =/= X ) ) -> ( ( T ` { Y , X } ) ` Y ) = X )
14	11, 13	syl 17	. 2 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ X =/= Y ) ) -> ( ( T ` { Y , X } ) ` Y ) = X )
15	3, 14	syl5eqr 2519	1 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ X =/= Y ) ) -> ( ( T ` { X , Y } ) ` Y ) = X )

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