Theorem pmtrrn2 15965

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 2442	. . . . . . 7 |- dom ( F \ _I ) = dom ( F \ _I )
4	1, 2, 3	pmtrfrn 15963	. . . . . 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 1002	. . . 4 |- ( F e. R -> dom ( F \ _I ) ~~ 2o )
7		en2 7547	. . . 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 1001	. . . . . . 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 3376	. . . . . . 7 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) C_ D <-> { x , y } C_ D ) )
13		breq1 4294	. . . . . . . 8 |- ( dom ( F \ _I ) = { x , y } -> ( dom ( F \ _I ) ~~ 2o <-> { x , y } ~~ 2o ) )
14		fveq2 5690	. . . . . . . . 9 |- ( dom ( F \ _I ) = { x , y } -> ( T ` dom ( F \ _I ) ) = ( T ` { x , y } ) )
15	14	eqeq2d 2453	. . . . . . . 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 2974	. . . . . . . 8 |- x e. _V
20		vex 2974	. . . . . . . 8 |- y e. _V
21	19, 20	prss 4026	. . . . . . 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 8171	. . . . . . . 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 1678	. . 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 2752	. 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 965   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2605   E.wrex 2715   _Vcvv 2971   \ cdif 3324   C_ wss 3327   {cpr 3878   class class class wbr 4291   _I cid 4630   dom cdm 4839   ran crn 4840   ` cfv 5417   2oc2o 6913   ~~ cen 7306  pmTrspcpmtr 15946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  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-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-om 6476  df-1o 6919  df-2o 6920  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-pmtr 15947

This theorem is referenced by:  mdetunilem7  18423