Theorem pmtrrn2 17179

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 2471	. . . . . . 7 |- dom ( F \ _I ) = dom ( F \ _I )
4	1, 2, 3	pmtrfrn 17177	. . . . . 6 |- ( F e. R -> ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) )
5	4	simpld 466	. . . . 5 |- ( F e. R -> ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) )
6	5	simp3d 1044	. . . 4 |- ( F e. R -> dom ( F \ _I ) ~~ 2o )
7		en2 7825	. . . 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 1043	. . . . . . 7 |- ( F e. R -> dom ( F \ _I ) C_ D )
10	4	simprd 470	. . . . . . 7 |- ( F e. R -> F = ( T ` dom ( F \ _I ) ) )
11	9, 6, 10	jca32 544	. . . . . 6 |- ( F e. R -> ( dom ( F \ _I ) C_ D /\ ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) ) )
12		sseq1 3439	. . . . . . 7 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) C_ D <-> { x , y } C_ D ) )
13		breq1 4398	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) ~~ 2o <-> { x , y } ~~ 2o ) )
14		fveq2 5879	. . . . . . . . 9 |- ( dom ( F \ _I ) = { x , y } -> ( T ` dom ( F \ _I ) ) = ( T ` { x , y } ) )
15	14	eqeq2d 2481	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( F = ( T ` dom ( F \ _I ) ) <-> F = ( T ` { x , y } ) ) )
16	13, 15	anbi12d 725	. . . . . . 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 725	. . . . . 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 228	. . . . 5 |- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) ) )
19		vex 3034	. . . . . . . 8 |- x e. _V
20		vex 3034	. . . . . . . 8 |- y e. _V
21	19, 20	prss 4117	. . . . . . 7 |- ( ( x e. D /\ y e. D ) <-> { x , y } C_ D )
22	21	bicomi 207	. . . . . 6 |- ( { x , y } C_ D <-> ( x e. D /\ y e. D ) )
23		pr2ne 8454	. . . . . . . 8 |- ( ( x e. _V /\ y e. _V ) -> ( { x , y } ~~ 2o <-> x =/= y ) )
24	19, 20, 23	mp2an 686	. . . . . . 7 |- ( { x , y } ~~ 2o <-> x =/= y )
25	24	anbi1i 709	. . . . . 6 |- ( ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) <-> ( x =/= y /\ F = ( T ` { x , y } ) ) )
26	22, 25	anbi12i 711	. . . . 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 234	. . . 4 |- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) )
28	27	2eximdv 1774	. . 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 2901	. 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 217	1 |- ( F e. R -> E. x e. D E. y e. D ( x =/= y /\ F = ( T ` { x , y } ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   E.wrex 2757   _Vcvv 3031   \ cdif 3387   C_ wss 3390   {cpr 3961   class class class wbr 4395   _I cid 4749   dom cdm 4839   ran crn 4840   ` cfv 5589   2oc2o 7194   ~~ cen 7584  pmTrspcpmtr 17160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-om 6712  df-1o 7200  df-2o 7201  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pmtr 17161

This theorem is referenced by:  mdetunilem7  19720