Theorem pmtrfrn 17878

Description: A transposition (as a kind of function) is the function transposing the two points it moves. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

Ref	Expression
pmtrrn.t	|- T = (pmTrsp ` D )
pmtrrn.r	|- R = ran T
pmtrfrn.p	|- P = dom ( F \ _I )

Assertion

Ref	Expression
pmtrfrn	|- ( F e. R -> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) )

Proof of Theorem pmtrfrn

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3919	. . . 4 |- -. F e. (/)
2		pmtrrn.r	. . . . . 6 |- R = ran T
3		pmtrrn.t	. . . . . . . . 9 |- T = (pmTrsp ` D )
4		fvprc 6185	. . . . . . . . 9 |- ( -. D e. _V -> (pmTrsp ` D ) = (/) )
5	3, 4	syl5eq 2668	. . . . . . . 8 |- ( -. D e. _V -> T = (/) )
6	5	rneqd 5353	. . . . . . 7 |- ( -. D e. _V -> ran T = ran (/) )
7		rn0 5377	. . . . . . 7 |- ran (/) = (/)
8	6, 7	syl6eq 2672	. . . . . 6 |- ( -. D e. _V -> ran T = (/) )
9	2, 8	syl5eq 2668	. . . . 5 |- ( -. D e. _V -> R = (/) )
10	9	eleq2d 2687	. . . 4 |- ( -. D e. _V -> ( F e. R <-> F e. (/) ) )
11	1, 10	mtbiri 317	. . 3 |- ( -. D e. _V -> -. F e. R )
12	11	con4i 113	. 2 |- ( F e. R -> D e. _V )
13		mptexg 6484	. . . . . . . 8 |- ( D e. _V -> ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) e. _V )
14	13	ralrimivw 2967	. . . . . . 7 |- ( D e. _V -> A. w e. { x e. ~P D | x ~~ 2o } ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) e. _V )
15		eqid 2622	. . . . . . . 8 |- ( w e. { x e. ~P D | x ~~ 2o } |-> ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) ) = ( w e. { x e. ~P D | x ~~ 2o } |-> ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) )
16	15	fnmpt 6020	. . . . . . 7 |- ( A. w e. { x e. ~P D | x ~~ 2o } ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) e. _V -> ( w e. { x e. ~P D | x ~~ 2o } |-> ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) ) Fn { x e. ~P D | x ~~ 2o } )
17	14, 16	syl 17	. . . . . 6 |- ( D e. _V -> ( w e. { x e. ~P D | x ~~ 2o } |-> ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) ) Fn { x e. ~P D | x ~~ 2o } )
18	3	pmtrfval 17870	. . . . . . 7 |- ( D e. _V -> T = ( w e. { x e. ~P D | x ~~ 2o } |-> ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) ) )
19	18	fneq1d 5981	. . . . . 6 |- ( D e. _V -> ( T Fn { x e. ~P D | x ~~ 2o } <-> ( w e. { x e. ~P D | x ~~ 2o } |-> ( z e. D |-> if ( z e. w , U. ( w \ { z } ) , z ) ) ) Fn { x e. ~P D | x ~~ 2o } ) )
20	17, 19	mpbird 247	. . . . 5 |- ( D e. _V -> T Fn { x e. ~P D | x ~~ 2o } )
21		fvelrnb 6243	. . . . 5 |- ( T Fn { x e. ~P D | x ~~ 2o } -> ( F e. ran T <-> E. y e. { x e. ~P D | x ~~ 2o } ( T ` y ) = F ) )
22	20, 21	syl 17	. . . 4 |- ( D e. _V -> ( F e. ran T <-> E. y e. { x e. ~P D | x ~~ 2o } ( T ` y ) = F ) )
23	2	eleq2i 2693	. . . 4 |- ( F e. R <-> F e. ran T )
24		breq1 4656	. . . . . 6 |- ( x = y -> ( x ~~ 2o <-> y ~~ 2o ) )
25	24	rexrab 3370	. . . . 5 |- ( E. y e. { x e. ~P D | x ~~ 2o } ( T ` y ) = F <-> E. y e. ~P D ( y ~~ 2o /\ ( T ` y ) = F ) )
26	25	bicomi 214	. . . 4 |- ( E. y e. ~P D ( y ~~ 2o /\ ( T ` y ) = F ) <-> E. y e. { x e. ~P D | x ~~ 2o } ( T ` y ) = F )
27	22, 23, 26	3bitr4g 303	. . 3 |- ( D e. _V -> ( F e. R <-> E. y e. ~P D ( y ~~ 2o /\ ( T ` y ) = F ) ) )
28		elpwi 4168	. . . . 5 |- ( y e. ~P D -> y C_ D )
29		simp1 1061	. . . . . . . . . 10 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> D e. _V )
30	3	pmtrmvd 17876	. . . . . . . . . . 11 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> dom ( ( T ` y ) \ _I ) = y )
31		simp2 1062	. . . . . . . . . . 11 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> y C_ D )
32	30, 31	eqsstrd 3639	. . . . . . . . . 10 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> dom ( ( T ` y ) \ _I ) C_ D )
33		simp3 1063	. . . . . . . . . . 11 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> y ~~ 2o )
34	30, 33	eqbrtrd 4675	. . . . . . . . . 10 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> dom ( ( T ` y ) \ _I ) ~~ 2o )
35	29, 32, 34	3jca 1242	. . . . . . . . 9 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> ( D e. _V /\ dom ( ( T ` y ) \ _I ) C_ D /\ dom ( ( T ` y ) \ _I ) ~~ 2o ) )
36	30	eqcomd 2628	. . . . . . . . . 10 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> y = dom ( ( T ` y ) \ _I ) )
37	36	fveq2d 6195	. . . . . . . . 9 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> ( T ` y ) = ( T ` dom ( ( T ` y ) \ _I ) ) )
38	35, 37	jca 554	. . . . . . . 8 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> ( ( D e. _V /\ dom ( ( T ` y ) \ _I ) C_ D /\ dom ( ( T ` y ) \ _I ) ~~ 2o ) /\ ( T ` y ) = ( T ` dom ( ( T ` y ) \ _I ) ) ) )
39		difeq1 3721	. . . . . . . . . . 11 |- ( ( T ` y ) = F -> ( ( T ` y ) \ _I ) = ( F \ _I ) )
40	39	dmeqd 5326	. . . . . . . . . 10 |- ( ( T ` y ) = F -> dom ( ( T ` y ) \ _I ) = dom ( F \ _I ) )
41		pmtrfrn.p	. . . . . . . . . 10 |- P = dom ( F \ _I )
42	40, 41	syl6eqr 2674	. . . . . . . . 9 |- ( ( T ` y ) = F -> dom ( ( T ` y ) \ _I ) = P )
43		sseq1 3626	. . . . . . . . . . . 12 |- ( dom ( ( T ` y ) \ _I ) = P -> ( dom ( ( T ` y ) \ _I ) C_ D <-> P C_ D ) )
44		breq1 4656	. . . . . . . . . . . 12 |- ( dom ( ( T ` y ) \ _I ) = P -> ( dom ( ( T ` y ) \ _I ) ~~ 2o <-> P ~~ 2o ) )
45	43, 44	3anbi23d 1402	. . . . . . . . . . 11 |- ( dom ( ( T ` y ) \ _I ) = P -> ( ( D e. _V /\ dom ( ( T ` y ) \ _I ) C_ D /\ dom ( ( T ` y ) \ _I ) ~~ 2o ) <-> ( D e. _V /\ P C_ D /\ P ~~ 2o ) ) )
46	45	adantl 482	. . . . . . . . . 10 |- ( ( ( T ` y ) = F /\ dom ( ( T ` y ) \ _I ) = P ) -> ( ( D e. _V /\ dom ( ( T ` y ) \ _I ) C_ D /\ dom ( ( T ` y ) \ _I ) ~~ 2o ) <-> ( D e. _V /\ P C_ D /\ P ~~ 2o ) ) )
47		simpl 473	. . . . . . . . . . 11 |- ( ( ( T ` y ) = F /\ dom ( ( T ` y ) \ _I ) = P ) -> ( T ` y ) = F )
48		fveq2 6191	. . . . . . . . . . . 12 |- ( dom ( ( T ` y ) \ _I ) = P -> ( T ` dom ( ( T ` y ) \ _I ) ) = ( T ` P ) )
49	48	adantl 482	. . . . . . . . . . 11 |- ( ( ( T ` y ) = F /\ dom ( ( T ` y ) \ _I ) = P ) -> ( T ` dom ( ( T ` y ) \ _I ) ) = ( T ` P ) )
50	47, 49	eqeq12d 2637	. . . . . . . . . 10 |- ( ( ( T ` y ) = F /\ dom ( ( T ` y ) \ _I ) = P ) -> ( ( T ` y ) = ( T ` dom ( ( T ` y ) \ _I ) ) <-> F = ( T ` P ) ) )
51	46, 50	anbi12d 747	. . . . . . . . 9 |- ( ( ( T ` y ) = F /\ dom ( ( T ` y ) \ _I ) = P ) -> ( ( ( D e. _V /\ dom ( ( T ` y ) \ _I ) C_ D /\ dom ( ( T ` y ) \ _I ) ~~ 2o ) /\ ( T ` y ) = ( T ` dom ( ( T ` y ) \ _I ) ) ) <-> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) ) )
52	42, 51	mpdan 702	. . . . . . . 8 |- ( ( T ` y ) = F -> ( ( ( D e. _V /\ dom ( ( T ` y ) \ _I ) C_ D /\ dom ( ( T ` y ) \ _I ) ~~ 2o ) /\ ( T ` y ) = ( T ` dom ( ( T ` y ) \ _I ) ) ) <-> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) ) )
53	38, 52	syl5ibcom 235	. . . . . . 7 |- ( ( D e. _V /\ y C_ D /\ y ~~ 2o ) -> ( ( T ` y ) = F -> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) ) )
54	53	3exp 1264	. . . . . 6 |- ( D e. _V -> ( y C_ D -> ( y ~~ 2o -> ( ( T ` y ) = F -> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) ) ) ) )
55	54	imp4a 614	. . . . 5 |- ( D e. _V -> ( y C_ D -> ( ( y ~~ 2o /\ ( T ` y ) = F ) -> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) ) ) )
56	28, 55	syl5 34	. . . 4 |- ( D e. _V -> ( y e. ~P D -> ( ( y ~~ 2o /\ ( T ` y ) = F ) -> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) ) ) )
57	56	rexlimdv 3030	. . 3 |- ( D e. _V -> ( E. y e. ~P D ( y ~~ 2o /\ ( T ` y ) = F ) -> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) ) )
58	27, 57	sylbid 230	. 2 |- ( D e. _V -> ( F e. R -> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) ) )
59	12, 58	mpcom 38	1 |- ( F e. R -> ( ( D e. _V /\ P C_ D /\ P ~~ 2o ) /\ F = ( T ` P ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   dom cdm 5114   ran crn 5115   Fn wfn 5883   ` 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-fin 7959  df-pmtr 17862

This theorem is referenced by:  pmtrffv  17879  pmtrrn2  17880  pmtrfinv  17881  pmtrfmvdn0  17882  pmtrff1o  17883  pmtrfcnv  17884  pmtrfb  17885