Theorem pmtrrn2 17880

Description: For any transposition there are two points it is transposing. (Contributed by SO, 15-Jul-2018.)

Hypotheses

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

Assertion

Ref	Expression
pmtrrn2	|- ( F e. R -> E. x e. D E. y e. D ( x =/= y /\ F = ( T ` { x , y } ) ) )

Distinct variable groups:   x, y, D   x, T, y   x, F, y   x, R, y

Proof of Theorem pmtrrn2

Step	Hyp	Ref	Expression
1		pmtrrn.t	. . . . . . 7 |- T = (pmTrsp ` D )
2		pmtrrn.r	. . . . . . 7 |- R = ran T
3		eqid 2622	. . . . . . 7 |- dom ( F \ _I ) = dom ( F \ _I )
4	1, 2, 3	pmtrfrn 17878	. . . . . 6 |- ( F e. R -> ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) )
5	4	simpld 475	. . . . 5 |- ( F e. R -> ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) )
6	5	simp3d 1075	. . . 4 |- ( F e. R -> dom ( F \ _I ) ~~ 2o )
7		en2 8196	. . . 4 |- ( dom ( F \ _I ) ~~ 2o -> E. x E. y dom ( F \ _I ) = { x , y } )
8	6, 7	syl 17	. . 3 |- ( F e. R -> E. x E. y dom ( F \ _I ) = { x , y } )
9	5	simp2d 1074	. . . . . . 7 |- ( F e. R -> dom ( F \ _I ) C_ D )
10	4	simprd 479	. . . . . . 7 |- ( F e. R -> F = ( T ` dom ( F \ _I ) ) )
11	9, 6, 10	jca32 558	. . . . . 6 |- ( F e. R -> ( dom ( F \ _I ) C_ D /\ ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) ) )
12		sseq1 3626	. . . . . . 7 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) C_ D <-> { x , y } C_ D ) )
13		breq1 4656	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) ~~ 2o <-> { x , y } ~~ 2o ) )
14		fveq2 6191	. . . . . . . . 9 |- ( dom ( F \ _I ) = { x , y } -> ( T ` dom ( F \ _I ) ) = ( T ` { x , y } ) )
15	14	eqeq2d 2632	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( F = ( T ` dom ( F \ _I ) ) <-> F = ( T ` { x , y } ) ) )
16	13, 15	anbi12d 747	. . . . . . 7 |- ( dom ( F \ _I ) = { x , y } -> ( ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) <-> ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) )
17	12, 16	anbi12d 747	. . . . . 6 |- ( dom ( F \ _I ) = { x , y } -> ( ( dom ( F \ _I ) C_ D /\ ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) ) <-> ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) ) )
18	11, 17	syl5ibcom 235	. . . . 5 |- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) ) )
19		vex 3203	. . . . . . . 8 |- x e. _V
20		vex 3203	. . . . . . . 8 |- y e. _V
21	19, 20	prss 4351	. . . . . . 7 |- ( ( x e. D /\ y e. D ) <-> { x , y } C_ D )
22	21	bicomi 214	. . . . . 6 |- ( { x , y } C_ D <-> ( x e. D /\ y e. D ) )
23		pr2ne 8828	. . . . . . . 8 |- ( ( x e. _V /\ y e. _V ) -> ( { x , y } ~~ 2o <-> x =/= y ) )
24	19, 20, 23	mp2an 708	. . . . . . 7 |- ( { x , y } ~~ 2o <-> x =/= y )
25	24	anbi1i 731	. . . . . 6 |- ( ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) <-> ( x =/= y /\ F = ( T ` { x , y } ) ) )
26	22, 25	anbi12i 733	. . . . 5 |- ( ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) <-> ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) )
27	18, 26	syl6ib 241	. . . 4 |- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) )
28	27	2eximdv 1848	. . 3 |- ( F e. R -> ( E. x E. y dom ( F \ _I ) = { x , y } -> E. x E. y ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) )
29	8, 28	mpd 15	. 2 |- ( F e. R -> E. x E. y ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) )
30		r2ex 3061	. 2 |- ( E. x e. D E. y e. D ( x =/= y /\ F = ( T ` { x , y } ) ) <-> E. x E. y ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) )
31	29, 30	sylibr 224	1 |- ( F e. R -> E. x e. D E. y e. D ( x =/= y /\ F = ( T ` { x , y } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {cpr 4179   class class class wbr 4653   _I cid 5023   dom cdm 5114   ran crn 5115   ` 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:  mdetunilem7  20424