Theorem pmtrrn2 17101

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 2451	. . . . . . 7 |- dom ( F \ _I ) = dom ( F \ _I )
4	1, 2, 3	pmtrfrn 17099	. . . . . 6 |- ( F e. R -> ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) )
5	4	simpld 461	. . . . 5 |- ( F e. R -> ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) )
6	5	simp3d 1022	. . . 4 |- ( F e. R -> dom ( F \ _I ) ~~ 2o )
7		en2 7807	. . . 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 1021	. . . . . . 7 |- ( F e. R -> dom ( F \ _I ) C_ D )
10	4	simprd 465	. . . . . . 7 |- ( F e. R -> F = ( T ` dom ( F \ _I ) ) )
11	9, 6, 10	jca32 538	. . . . . 6 |- ( F e. R -> ( dom ( F \ _I ) C_ D /\ ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) ) )
12		sseq1 3453	. . . . . . 7 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) C_ D <-> { x , y } C_ D ) )
13		breq1 4405	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) ~~ 2o <-> { x , y } ~~ 2o ) )
14		fveq2 5865	. . . . . . . . 9 |- ( dom ( F \ _I ) = { x , y } -> ( T ` dom ( F \ _I ) ) = ( T ` { x , y } ) )
15	14	eqeq2d 2461	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( F = ( T ` dom ( F \ _I ) ) <-> F = ( T ` { x , y } ) ) )
16	13, 15	anbi12d 717	. . . . . . 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 717	. . . . . 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 224	. . . . 5 |- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) ) )
19		vex 3048	. . . . . . . 8 |- x e. _V
20		vex 3048	. . . . . . . 8 |- y e. _V
21	19, 20	prss 4126	. . . . . . 7 |- ( ( x e. D /\ y e. D ) <-> { x , y } C_ D )
22	21	bicomi 206	. . . . . 6 |- ( { x , y } C_ D <-> ( x e. D /\ y e. D ) )
23		pr2ne 8436	. . . . . . . 8 |- ( ( x e. _V /\ y e. _V ) -> ( { x , y } ~~ 2o <-> x =/= y ) )
24	19, 20, 23	mp2an 678	. . . . . . 7 |- ( { x , y } ~~ 2o <-> x =/= y )
25	24	anbi1i 701	. . . . . 6 |- ( ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) <-> ( x =/= y /\ F = ( T ` { x , y } ) ) )
26	22, 25	anbi12i 703	. . . . 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 230	. . . 4 |- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) )
28	27	2eximdv 1766	. . 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 2913	. 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 216	1 |- ( F e. R -> E. x e. D E. y e. D ( x =/= y /\ F = ( T ` { x , y } ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   E.wrex 2738   _Vcvv 3045   \ cdif 3401   C_ wss 3404   {cpr 3970   class class class wbr 4402   _I cid 4744   dom cdm 4834   ran crn 4835   ` cfv 5582   2oc2o 7176   ~~ cen 7566  pmTrspcpmtr 17082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-om 6693  df-1o 7182  df-2o 7183  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pmtr 17083

This theorem is referenced by:  mdetunilem7  19643