Theorem pmtrrn2 16359

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 2443	. . . . . . 7 |- dom ( F \ _I ) = dom ( F \ _I )
4	1, 2, 3	pmtrfrn 16357	. . . . . 6 |- ( F e. R -> ( ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) /\ F = ( T ` dom ( F \ _I ) ) ) )
5	4	simpld 459	. . . . 5 |- ( F e. R -> ( D e. _V /\ dom ( F \ _I ) C_ D /\ dom ( F \ _I ) ~~ 2o ) )
6	5	simp3d 1011	. . . 4 |- ( F e. R -> dom ( F \ _I ) ~~ 2o )
7		en2 7758	. . . 4 |- ( dom ( F \ _I ) ~~ 2o -> E. x E. y dom ( F \ _I ) = { x , y } )
8	6, 7	syl 16	. . 3 |- ( F e. R -> E. x E. y dom ( F \ _I ) = { x , y } )
9	5	simp2d 1010	. . . . . . 7 |- ( F e. R -> dom ( F \ _I ) C_ D )
10	4	simprd 463	. . . . . . 7 |- ( F e. R -> F = ( T ` dom ( F \ _I ) ) )
11	9, 6, 10	jca32 535	. . . . . 6 |- ( F e. R -> ( dom ( F \ _I ) C_ D /\ ( dom ( F \ _I ) ~~ 2o /\ F = ( T ` dom ( F \ _I ) ) ) ) )
12		sseq1 3510	. . . . . . 7 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) C_ D <-> { x , y } C_ D ) )
13		breq1 4440	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) ~~ 2o <-> { x , y } ~~ 2o ) )
14		fveq2 5856	. . . . . . . . 9 |- ( dom ( F \ _I ) = { x , y } -> ( T ` dom ( F \ _I ) ) = ( T ` { x , y } ) )
15	14	eqeq2d 2457	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( F = ( T ` dom ( F \ _I ) ) <-> F = ( T ` { x , y } ) ) )
16	13, 15	anbi12d 710	. . . . . . 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 710	. . . . . 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 220	. . . . 5 |- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( { x , y } C_ D /\ ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) ) ) )
19		vex 3098	. . . . . . . 8 |- x e. _V
20		vex 3098	. . . . . . . 8 |- y e. _V
21	19, 20	prss 4169	. . . . . . 7 |- ( ( x e. D /\ y e. D ) <-> { x , y } C_ D )
22	21	bicomi 202	. . . . . 6 |- ( { x , y } C_ D <-> ( x e. D /\ y e. D ) )
23		pr2ne 8386	. . . . . . . 8 |- ( ( x e. _V /\ y e. _V ) -> ( { x , y } ~~ 2o <-> x =/= y ) )
24	19, 20, 23	mp2an 672	. . . . . . 7 |- ( { x , y } ~~ 2o <-> x =/= y )
25	24	anbi1i 695	. . . . . 6 |- ( ( { x , y } ~~ 2o /\ F = ( T ` { x , y } ) ) <-> ( x =/= y /\ F = ( T ` { x , y } ) ) )
26	22, 25	anbi12i 697	. . . . 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 226	. . . 4 |- ( F e. R -> ( dom ( F \ _I ) = { x , y } -> ( ( x e. D /\ y e. D ) /\ ( x =/= y /\ F = ( T ` { x , y } ) ) ) ) )
28	27	2eximdv 1699	. . 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 2966	. 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 212	1 |- ( F e. R -> E. x e. D E. y e. D ( x =/= y /\ F = ( T ` { x , y } ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   E.wex 1599   e. wcel 1804   =/= wne 2638   E.wrex 2794   _Vcvv 3095   \ cdif 3458   C_ wss 3461   {cpr 4016   class class class wbr 4437   _I cid 4780   dom cdm 4989   ran crn 4990   ` cfv 5578   2oc2o 7126   ~~ cen 7515  pmTrspcpmtr 16340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-om 6686  df-1o 7132  df-2o 7133  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pmtr 16341

This theorem is referenced by:  mdetunilem7  18993