Theorem grplactf1o 9406

Description: The left group action of element A of group G maps the underlying set X of G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.)

Hypotheses

Ref	Expression
grplact.1	|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
grplact.2	|- X = ran G

Assertion

Ref	Expression
grplactf1o	|- ((G e. Grp /\ A e. X) -> (F` A):X-1-1-onto->X)

Distinct variable groups:   A,a,b,g,h   G,a,b,g,h   X,a,b,g,h

Proof of Theorem grplactf1o

Step	Hyp	Ref	Expression
1		dff1o5 4646	. 2 |- ((F` A):X-1-1-onto->X <-> ((F` A):X-1-1->X /\ ran ( F` A) = X))
2		dff13 4850	. . 3 |- ((F` A):X-1-1->X <-> ((F` A):X-->X /\ A.x e. X A.y e. X (((F` A)` x) = ((F` A)` y) -> x = y)))
3		grplact.2	. . . . . . . 8 |- X = ran G
4	3	grpcl 9324	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ a e. X) -> (AGa) e. X)
5	4	3expia 1069	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (a e. X -> (AGa) e. X))
6	5	r19.21aiv 2175	. . . . 5 |- ((G e. Grp /\ A e. X) -> A.a e. X (AGa) e. X)
7		eqid 1884	. . . . . 6 |- {<.a, b>. | (a e. X /\ b = (AGa))} = {<.a, b>. | (a e. X /\ b = (AGa))}
8	7	fopab2 4796	. . . . 5 |- (A.a e. X (AGa) e. X <-> {<.a, b>. | (a e. X /\ b = (AGa))}:X-->X)
9	6, 8	sylib 215	. . . 4 |- ((G e. Grp /\ A e. X) -> {<.a, b>. | (a e. X /\ b = (AGa))}:X-->X)
10		grplact.1	. . . . . 6 |- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
11	10, 3	grplactfval 9404	. . . . 5 |- ((G e. Grp /\ A e. X) -> (F` A) = {<.a, b>. | (a e. X /\ b = (AGa))})
12	11	feq1d 4556	. . . 4 |- ((G e. Grp /\ A e. X) -> ((F` A):X-->X <-> {<.a, b>. | (a e. X /\ b = (AGa))}:X-->X))
13	9, 12	mpbird 213	. . 3 |- ((G e. Grp /\ A e. X) -> (F` A):X-->X)
14	10, 3	grplactval 9405	. . . . . . . . . . 11 |- ((G e. Grp /\ A e. X /\ x e. X) -> ((F` A)` x) = (AGx))
15	14	3expia 1069	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X) -> (x e. X -> ((F` A)` x) = (AGx)))
16	10, 3	grplactval 9405	. . . . . . . . . . 11 |- ((G e. Grp /\ A e. X /\ y e. X) -> ((F` A)` y) = (AGy))
17	16	3expia 1069	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X) -> (y e. X -> ((F` A)` y) = (AGy)))
18	15, 17	anim12d 617	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> ((x e. X /\ y e. X) -> (((F` A)` x) = (AGx) /\ ((F` A)` y) = (AGy))))
19	18	imp 377	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ (x e. X /\ y e. X)) -> (((F` A)` x) = (AGx) /\ ((F` A)` y) = (AGy)))
20		eqeq12 1896	. . . . . . . 8 |- ((((F` A)` x) = (AGx) /\ ((F` A)` y) = (AGy)) -> (((F` A)` x) = ((F` A)` y) <-> (AGx) = (AGy)))
21	19, 20	syl 12	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ (x e. X /\ y e. X)) -> (((F` A)` x) = ((F` A)` y) <-> (AGx) = (AGy)))
22	3	grplcan 9359	. . . . . . . . . . . 12 |- ((G e. Grp /\ (x e. X /\ y e. X /\ A e. X)) -> ((AGx) = (AGy) <-> x = y))
23	22	expcom 403	. . . . . . . . . . 11 |- ((x e. X /\ y e. X /\ A e. X) -> (G e. Grp -> ((AGx) = (AGy) <-> x = y)))
24	23	3expia 1069	. . . . . . . . . 10 |- ((x e. X /\ y e. X) -> (A e. X -> (G e. Grp -> ((AGx) = (AGy) <-> x = y))))
25	24	com23 36	. . . . . . . . 9 |- ((x e. X /\ y e. X) -> (G e. Grp -> (A e. X -> ((AGx) = (AGy) <-> x = y))))
26	25	imp3a 388	. . . . . . . 8 |- ((x e. X /\ y e. X) -> ((G e. Grp /\ A e. X) -> ((AGx) = (AGy) <-> x = y)))
27	26	impcom 378	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ (x e. X /\ y e. X)) -> ((AGx) = (AGy) <-> x = y))
28	21, 27	bitrd 587	. . . . . 6 |- (((G e. Grp /\ A e. X) /\ (x e. X /\ y e. X)) -> (((F` A)` x) = ((F` A)` y) <-> x = y))
29	28	biimpd 170	. . . . 5 |- (((G e. Grp /\ A e. X) /\ (x e. X /\ y e. X)) -> (((F` A)` x) = ((F` A)` y) -> x = y))
30	29	ex 402	. . . 4 |- ((G e. Grp /\ A e. X) -> ((x e. X /\ y e. X) -> (((F` A)` x) = ((F` A)` y) -> x = y)))
31	30	r19.21aivv 2183	. . 3 |- ((G e. Grp /\ A e. X) -> A.x e. X A.y e. X (((F` A)` x) = ((F` A)` y) -> x = y))
32	2, 13, 31	sylanbrc 527	. 2 |- ((G e. Grp /\ A e. X) -> (F` A):X-1-1->X)
33		frn 4569	. . . 4 |- ((F` A):X-->X -> ran ( F` A) C_ X)
34	13, 33	syl 12	. . 3 |- ((G e. Grp /\ A e. X) -> ran ( F` A) C_ X)
35		eqid 1884	. . . . . . . . . 10 |- (inv` G) = (inv` G)
36	3, 35	grpinvcl 9352	. . . . . . . . 9 |- ((G e. Grp /\ A e. X) -> ((inv` G)` A) e. X)
37	3	grpcl 9324	. . . . . . . . . 10 |- ((G e. Grp /\ ((inv` G)` A) e. X /\ y e. X) -> (((inv` G)` A)Gy) e. X)
38	37	3expia 1069	. . . . . . . . 9 |- ((G e. Grp /\ ((inv` G)` A) e. X) -> (y e. X -> (((inv` G)` A)Gy) e. X))
39	36, 38	syldan 516	. . . . . . . 8 |- ((G e. Grp /\ A e. X) -> (y e. X -> (((inv` G)` A)Gy) e. X))
40	39	3impia 1064	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ y e. X) -> (((inv` G)` A)Gy) e. X)
41	3, 35	grpasscan1 9361	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ y e. X) -> (AG(((inv` G)` A)Gy)) = y)
42	41	eqcomd 1889	. . . . . . 7 |- ((G e. Grp /\ A e. X /\ y e. X) -> y = (AG(((inv` G)` A)Gy)))
43		opreq2 4890	. . . . . . . . 9 |- (a = (((inv` G)` A)Gy) -> (AGa) = (AG(((inv` G)` A)Gy)))
44	43	eqeq2d 1895	. . . . . . . 8 |- (a = (((inv` G)` A)Gy) -> (y = (AGa) <-> y = (AG(((inv` G)` A)Gy))))
45	44	rcla4ev 2381	. . . . . . 7 |- (((((inv` G)` A)Gy) e. X /\ y = (AG(((inv` G)` A)Gy))) -> E.a e. X y = (AGa))
46	40, 42, 45	syl11anc 524	. . . . . 6 |- ((G e. Grp /\ A e. X /\ y e. X) -> E.a e. X y = (AGa))
47	46	3expia 1069	. . . . 5 |- ((G e. Grp /\ A e. X) -> (y e. X -> E.a e. X y = (AGa)))
48	11	rneqd 4188	. . . . . . 7 |- ((G e. Grp /\ A e. X) -> ran ( F` A) = ran {<.a, b>. | (a e. X /\ b = (AGa))})
49	48	eleq2d 1964	. . . . . 6 |- ((G e. Grp /\ A e. X) -> (y e. ran ( F` A) <-> y e. ran {<.a, b>. | (a e. X /\ b = (AGa))}))
50		oprex 4907	. . . . . . . 8 |- (AGa) e. _V
51	50, 7	elrnopab 4774	. . . . . . 7 |- (y e. ran {<.a, b>. | (a e. X /\ b = (AGa))} <-> E.a e. X y = (AGa))
52	51	biimpri 169	. . . . . 6 |- (E.a e. X y = (AGa) -> y e. ran {<.a, b>. | (a e. X /\ b = (AGa))})
53	49, 52	syl5bir 227	. . . . 5 |- ((G e. Grp /\ A e. X) -> (E.a e. X y = (AGa) -> y e. ran ( F` A)))
54	47, 53	syld 30	. . . 4 |- ((G e. Grp /\ A e. X) -> (y e. X -> y e. ran ( F` A)))
55	54	ssrdv 2622	. . 3 |- ((G e. Grp /\ A e. X) -> X C_ ran ( F` A))
56	34, 55	eqssd 2633	. 2 |- ((G e. Grp /\ A e. X) -> ran ( F` A) = X)
57	1, 32, 56	sylanbrc 527	1 |- ((G e. Grp /\ A e. X) -> (F` A):X-1-1-onto->X)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  {copab 3395  ran crn 3987  -->wf 3994  -1-1->wf1 3995  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  invcgn 9313

This theorem is referenced by:  shftefif1olem 10095  cayleylem1 13641

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-grp 9316  df-gid 9317  df-ginv 9318

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