Theorem pmtrfb 27406

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 2283	. . . . 5 |- dom ( F \ _I ) = dom ( F \ _I )
4	1, 2, 3	pmtrfrn 27400	. . . 4 |- ( F e. R -> ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) )
5		simpl1 958	. . . 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 15	. . 3 |- ( F e. R -> D e. _V )
7	1, 2	pmtrff1o 27404	. . 3 |- ( F e. R -> F : D -1-1-onto-> D )
8		simpl3 960	. . . 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 15	. . 3 |- ( F e. R -> dom ( F \ _I ) ~~ 2o )
10	6, 7, 9	3jca 1132	. 2 |- ( F e. R -> ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) )
11		simp2 956	. . . 4 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> F : D -1-1-onto-> D )
12		difss 3303	. . . . . . . 8 |- ( F \ _I ) C_ F
13		dmss 4878	. . . . . . . 8 |- ( ( F \ _I ) C_ F -> dom ( F \ _I ) C_ dom F )
14	12, 13	ax-mp 8	. . . . . . 7 |- dom ( F \ _I ) C_ dom F
15		f1odm 5476	. . . . . . 7 |- ( F : D -1-1-onto-> D -> dom F = D )
16	14, 15	syl5sseq 3226	. . . . . 6 |- ( F : D -1-1-onto-> D -> dom ( F \ _I ) C_ D )
17	1, 2	pmtrrn 27399	. . . . . 6 |- ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) -> ( T ` dom ( F \ _I ) ) e. R )
18	16, 17	syl3an2 1216	. . . . 5 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> ( T ` dom ( F \ _I ) ) e. R )
19	1, 2	pmtrff1o 27404	. . . . 5 |- ( ( T ` dom ( F \ _I ) ) e. R -> ( T ` dom ( F \ _I ) ) : D -1-1-onto-> D )
20	18, 19	syl 15	. . . 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 957	. . . 4 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> dom ( F \ _I ) ~~ 2o )
22	1	pmtrmvd 27398	. . . . 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 1216	. . . 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 27390	. . . 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 1183	. . 3 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> F = ( T ` dom ( F \ _I ) ) )
26	25, 18	eqeltrd 2357	. 2 |- ( ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) -> F e. R )
27	10, 26	impbii 180	1 |- ( F e. R <-> ( D e. _V /\ F : D -1-1-onto-> D /\ dom ( F \ _I ) ~~ 2o ) )

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934   = wceq 1623   e. wcel 1684   _Vcvv 2788   \ cdif 3149   C_ wss 3152   class class class wbr 4023   _I cid 4304   dom cdm 4689   ran crn 4690   -1-1-onto->wf1o 5254   ` cfv 5255   2oc2o 6473   ~~ cen 6860  pmTrspcpmtr 27384

This theorem is referenced by:  pmtrfconj  27407  symggen  27411  psgnunilem1  27416

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-2o 6480  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pmtr 27385