Theorem pmtrmvd 17876

Description: A transposition moves precisely the transposed points. (Contributed by Stefan O'Rear, 16-Aug-2015.)

Hypothesis

Ref	Expression
pmtrfval.t	|- T = (pmTrsp ` D )

Assertion

Ref	Expression
pmtrmvd	|- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> dom ( ( T ` P ) \ _I ) = P )

Proof of Theorem pmtrmvd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmtrfval.t	. . . 4 |- T = (pmTrsp ` D )
2	1	pmtrf 17875	. . 3 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( T ` P ) : D --> D )
3		ffn 6045	. . 3 |- ( ( T ` P ) : D --> D -> ( T ` P ) Fn D )
4		fndifnfp 6442	. . 3 |- ( ( T ` P ) Fn D -> dom ( ( T ` P ) \ _I ) = { z e. D | ( ( T ` P ) ` z ) =/= z } )
5	2, 3, 4	3syl 18	. 2 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> dom ( ( T ` P ) \ _I ) = { z e. D | ( ( T ` P ) ` z ) =/= z } )
6	1	pmtrfv 17872	. . . . . 6 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. D ) -> ( ( T ` P ) ` z ) = if ( z e. P , U. ( P \ { z } ) , z ) )
7	6	neeq1d 2853	. . . . 5 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. D ) -> ( ( ( T ` P ) ` z ) =/= z <-> if ( z e. P , U. ( P \ { z } ) , z ) =/= z ) )
8		iffalse 4095	. . . . . . . 8 |- ( -. z e. P -> if ( z e. P , U. ( P \ { z } ) , z ) = z )
9	8	necon1ai 2821	. . . . . . 7 |- ( if ( z e. P , U. ( P \ { z } ) , z ) =/= z -> z e. P )
10		iftrue 4092	. . . . . . . . . 10 |- ( z e. P -> if ( z e. P , U. ( P \ { z } ) , z ) = U. ( P \ { z } ) )
11	10	adantl 482	. . . . . . . . 9 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. P ) -> if ( z e. P , U. ( P \ { z } ) , z ) = U. ( P \ { z } ) )
12		1onn 7719	. . . . . . . . . . . 12 |- 1o e. om
13	12	a1i 11	. . . . . . . . . . 11 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. P ) -> 1o e. om )
14		simpl3 1066	. . . . . . . . . . . 12 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. P ) -> P ~~ 2o )
15		df-2o 7561	. . . . . . . . . . . 12 |- 2o = suc 1o
16	14, 15	syl6breq 4694	. . . . . . . . . . 11 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. P ) -> P ~~ suc 1o )
17		simpr 477	. . . . . . . . . . 11 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. P ) -> z e. P )
18		dif1en 8193	. . . . . . . . . . 11 |- ( ( 1o e. om /\ P ~~ suc 1o /\ z e. P ) -> ( P \ { z } ) ~~ 1o )
19	13, 16, 17, 18	syl3anc 1326	. . . . . . . . . 10 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. P ) -> ( P \ { z } ) ~~ 1o )
20		en1uniel 8028	. . . . . . . . . 10 |- ( ( P \ { z } ) ~~ 1o -> U. ( P \ { z } ) e. ( P \ { z } ) )
21		eldifsni 4320	. . . . . . . . . 10 |- ( U. ( P \ { z } ) e. ( P \ { z } ) -> U. ( P \ { z } ) =/= z )
22	19, 20, 21	3syl 18	. . . . . . . . 9 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. P ) -> U. ( P \ { z } ) =/= z )
23	11, 22	eqnetrd 2861	. . . . . . . 8 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. P ) -> if ( z e. P , U. ( P \ { z } ) , z ) =/= z )
24	23	ex 450	. . . . . . 7 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( z e. P -> if ( z e. P , U. ( P \ { z } ) , z ) =/= z ) )
25	9, 24	impbid2 216	. . . . . 6 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( if ( z e. P , U. ( P \ { z } ) , z ) =/= z <-> z e. P ) )
26	25	adantr 481	. . . . 5 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. D ) -> ( if ( z e. P , U. ( P \ { z } ) , z ) =/= z <-> z e. P ) )
27	7, 26	bitrd 268	. . . 4 |- ( ( ( D e. V /\ P C_ D /\ P ~~ 2o ) /\ z e. D ) -> ( ( ( T ` P ) ` z ) =/= z <-> z e. P ) )
28	27	rabbidva 3188	. . 3 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> { z e. D | ( ( T ` P ) ` z ) =/= z } = { z e. D | z e. P } )
29		incom 3805	. . . 4 |- ( P i^i D ) = ( D i^i P )
30		dfin5 3582	. . . 4 |- ( D i^i P ) = { z e. D | z e. P }
31	29, 30	eqtri 2644	. . 3 |- ( P i^i D ) = { z e. D | z e. P }
32	28, 31	syl6eqr 2674	. 2 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> { z e. D | ( ( T ` P ) ` z ) =/= z } = ( P i^i D ) )
33		simp2 1062	. . 3 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> P C_ D )
34		df-ss 3588	. . 3 |- ( P C_ D <-> ( P i^i D ) = P )
35	33, 34	sylib 208	. 2 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> ( P i^i D ) = P )
36	5, 32, 35	3eqtrd 2660	1 |- ( ( D e. V /\ P C_ D /\ P ~~ 2o ) -> dom ( ( T ` P ) \ _I ) = P )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   {csn 4177   U.cuni 4436   class class class wbr 4653   _I cid 5023   dom cdm 5114   suc csuc 5725   Fn wfn 5883   -->wf 5884   ` cfv 5888   omcom 7065   1oc1o 7553   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-fin 7959  df-pmtr 17862

This theorem is referenced by:  pmtrfrn  17878  pmtrfb  17885  symggen  17890  pmtrdifellem2  17897  mdetralt  20414  mdetunilem7  20424