Theorem pmtrprfv3 17874

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

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> D e. V )
2		simp1 1061	. . . . 5 |- ( ( X e. D /\ Y e. D /\ Z e. D ) -> X e. D )
3	2	3ad2ant2 1083	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> X e. D )
4		simp22 1095	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Y e. D )
5		prssi 4353	. . . 4 |- ( ( X e. D /\ Y e. D ) -> { X , Y } C_ D )
6	3, 4, 5	syl2anc 693	. . 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 8827	. . . . . . . . 9 |- ( ( X e. D /\ Y e. D /\ X =/= Y ) -> { X , Y } ~~ 2o )
8	7	3expia 1267	. . . . . . . 8 |- ( ( X e. D /\ Y e. D ) -> ( X =/= Y -> { X , Y } ~~ 2o ) )
9	8	3adant3 1081	. . . . . . 7 |- ( ( X e. D /\ Y e. D /\ Z e. D ) -> ( X =/= Y -> { X , Y } ~~ 2o ) )
10	9	com12 32	. . . . . 6 |- ( X =/= Y -> ( ( X e. D /\ Y e. D /\ Z e. D ) -> { X , Y } ~~ 2o ) )
11	10	3ad2ant1 1082	. . . . 5 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> ( ( X e. D /\ Y e. D /\ Z e. D ) -> { X , Y } ~~ 2o ) )
12	11	impcom 446	. . . 4 |- ( ( ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { X , Y } ~~ 2o )
13	12	3adant1 1079	. . 3 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { X , Y } ~~ 2o )
14		simp23 1096	. . 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 )
16	15	pmtrfv 17872	. . 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 ) )
17	1, 6, 13, 14, 16	syl31anc 1329	. 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 2847	. . . . . . 7 |- ( X =/= Z <-> Z =/= X )
19	18	biimpi 206	. . . . . 6 |- ( X =/= Z -> Z =/= X )
20	19	3ad2ant2 1083	. . . . 5 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> Z =/= X )
21	20	3ad2ant3 1084	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Z =/= X )
22		necom 2847	. . . . . . 7 |- ( Y =/= Z <-> Z =/= Y )
23	22	biimpi 206	. . . . . 6 |- ( Y =/= Z -> Z =/= Y )
24	23	3ad2ant3 1084	. . . . 5 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> Z =/= Y )
25	24	3ad2ant3 1084	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Z =/= Y )
26	21, 25	nelprd 4203	. . 3 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> -. Z e. { X , Y } )
27	26	iffalsed 4097	. 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 )
28	17, 27	eqtrd 2656	1 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( T ` { X , Y } ) ` Z ) = Z )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   ifcif 4086   {csn 4177   {cpr 4179   U.cuni 4436   class class class wbr 4653   ` cfv 5888   2oc2o 7554   ~~ cen 7952  pmTrspcpmtr 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pmtr 17862

This theorem is referenced by:  pmtr3ncomlem1  17893  psgnfzto1stlem  29850