Theorem pmtrfb 17885

Description: An intrinsic characterization of the transposition permutations. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Hypotheses

Ref	Expression
pmtrrn.t	|- T = (pmTrsp ` D )
pmtrrn.r	|- R = ran T

Assertion

Ref	Expression
pmtrfb	|- ( F e. R <-> ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) )

Proof of Theorem pmtrfb

Step	Hyp	Ref	Expression
1		pmtrrn.t	. . . . 5 |- T = (pmTrsp ` D )
2		pmtrrn.r	. . . . 5 |- R = ran T
3		eqid 2622	. . . . 5 |- dom ( F \ _I ) = dom ( F \ _I )
4	1, 2, 3	pmtrfrn 17878	. . . 4 |- ( F e. R -> ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) )
5		simpl1 1064	. . . 4 |- ( ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) -> D e. _V )
6	4, 5	syl 17	. . 3 |- ( F e. R -> D e. _V )
7	1, 2	pmtrff1o 17883	. . 3 |- ( F e. R -> F : D -1-1-onto-> D )
8		simpl3 1066	. . . 4 |- ( ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) -> dom ( F \ _I ) ~~ 2o )
9	4, 8	syl 17	. . 3 |- ( F e. R -> dom ( F \ _I ) ~~ 2o )
10	6, 7, 9	3jca 1242	. 2 |- ( F e. R -> ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) )
11		simp2 1062	. . . 4 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> F : D -1-1-onto-> D )
12		difss 3737	. . . . . . . 8 |- ( F \ _I ) C_ F
13		dmss 5323	. . . . . . . 8 |- ( ( F \ _I ) C_ F -> dom ( F \ _I ) C_ dom F )
14	12, 13	ax-mp 5	. . . . . . 7 |- dom ( F \ _I ) C_ dom F
15		f1odm 6141	. . . . . . 7 |- ( F : D -1-1-onto-> D -> dom F = D )
16	14, 15	syl5sseq 3653	. . . . . 6 |- ( F : D -1-1-onto-> D -> dom ( F \ _I ) C_ D )
17	1, 2	pmtrrn 17877	. . . . . 6 |- ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) -> ( T ` dom ( F \ _I ) ) e. R )
18	16, 17	syl3an2 1360	. . . . 5 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> ( T ` dom ( F \ _I ) ) e. R )
19	1, 2	pmtrff1o 17883	. . . . 5 |- ( ( T ` dom ( F \ _I ) ) e. R -> ( T ` dom ( F \ _I ) ) : D -1-1-onto-> D )
20	18, 19	syl 17	. . . 4 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> ( T ` dom ( F \ _I ) ) : D -1-1-onto-> D )
21		simp3 1063	. . . 4 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> dom ( F \ _I ) ~~ 2o )
22	1	pmtrmvd 17876	. . . . 5 |- ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) -> dom ( ( T ` dom ( F \ _I ) ) \ _I ) = dom ( F \ _I ) )
23	16, 22	syl3an2 1360	. . . 4 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> dom ( ( T ` dom ( F \ _I ) ) \ _I ) = dom ( F \ _I ) )
24		f1otrspeq 17867	. . . 4 |- ( ( ( F : D -1-1-onto-> D /\ ( T ` dom ( F \ _I ) ) : D -1-1-onto-> D ) /\ ( dom ( F \ _I ) ~~ 2o /\ dom ( ( T ` dom ( F \ _I ) ) \ _I ) = dom ( F \ _I ) ) ) -> F = ( T ` dom ( F \ _I ) ) )
25	11, 20, 21, 23, 24	syl22anc 1327	. . 3 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> F = ( T ` dom ( F \ _I ) ) )
26	25, 18	eqeltrd 2701	. 2 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> F e. R )
27	10, 26	impbii 199	1 |- ( F e. R <-> ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   \ cdif 3571   C_ wss 3574   class class class wbr 4653   _I cid 5023   dom cdm 5114   ran crn 5115   -1-1-onto->wf1o 5887   ` 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-fin 7959  df-pmtr 17862

This theorem is referenced by:  pmtrfconj  17886  symggen  17890  pmtrdifellem1  17896  pmtrdifellem2  17897  psgnunilem1  17913