Theorem cayleylem3 13643

Description: Lemma for cayleyi 13644.

Hypotheses

Ref	Expression
cayleylem1.1	|- G e. Grp
cayleylem1.2	|- P = {f | f:X-1-1-onto->X}
cayleylem1.3	|- X = ran G
cayleylem1.4	|- U = (Id` G)
cayleylem1.5	|- H = (SymGrp` X)
cayleylem1.6	|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
cayleylem1.7	|- Y = ran F
cayleylem1.8	|- S = (H |` (Y X. Y))

Assertion

Ref	Expression
cayleylem3	|- (S e. (SubGrp` H) /\ F e. (G GrpIso S))

Distinct variable groups:   a,b,f,g,h   f,F   G,a,b,f,g,h   P,g,h   U,a,b   X,a,b,f,g,h

Proof of Theorem cayleylem3

Step	Hyp	Ref	Expression
1		cayleylem1.1	. . 3 |- G e. Grp
2		cayleylem1.5	. . . 4 |- H = (SymGrp` X)
3		cayleylem1.3	. . . . . 6 |- X = ran G
4		rnexg 4207	. . . . . . 7 |- (G e. Grp -> ran G e. _V)
5	1, 4	ax-mp 7	. . . . . 6 |- ran G e. _V
6	3, 5	eqeltri 1967	. . . . 5 |- X e. _V
7	6	symggrpi 10205	. . . 4 |- (SymGrp` X) e. Grp
8	2, 7	eqeltri 1967	. . 3 |- H e. Grp
9		cayleylem1.2	. . . 4 |- P = {f | f:X-1-1-onto->X}
10		cayleylem1.4	. . . 4 |- U = (Id` G)
11		cayleylem1.6	. . . 4 |- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
12		cayleylem1.7	. . . 4 |- Y = ran F
13		cayleylem1.8	. . . 4 |- S = (H |` (Y X. Y))
14	1, 9, 3, 10, 2, 11, 12, 13	cayleylem2 13642	. . 3 |- F e. (G GrpHom H)
15	1, 8, 14, 12, 13	ghomgrpi 13630	. 2 |- S e. (SubGrp` H)
16		issubg 9425	. . . . . 6 |- (S e. (SubGrp` H) <-> (H e. Grp /\ S e. Grp /\ S C_ H))
17	15, 16	mpbi 206	. . . . 5 |- (H e. Grp /\ S e. Grp /\ S C_ H)
18	17	simp2i 886	. . . 4 |- S e. Grp
19		elgiso 13639	. . . 4 |- ((G e. Grp /\ S e. Grp) -> (F e. (G GrpIso S) <-> (F e. (G GrpHom S) /\ F:ran G-1-1-onto->ran S)))
20	1, 18, 19	mp2an 761	. . 3 |- (F e. (G GrpIso S) <-> (F e. (G GrpHom S) /\ F:ran G-1-1-onto->ran S))
21	12, 13	ghomgsg 13636	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F e. (G GrpHom S))
22	1, 8, 14, 21	mp3an 1191	. . 3 |- F e. (G GrpHom S)
23		eqid 1884	. . . . . . . . . . . . . . . 16 |- ran S = ran S
24	3, 12, 13, 23	ghomfo 13634	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-onto->ran S)
25		forn 4620	. . . . . . . . . . . . . . 15 |- (F:X-onto->ran S -> ran F = ran S)
26	24, 25	syl 12	. . . . . . . . . . . . . 14 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ran F = ran S)
27	1, 8, 14, 26	mp3an 1191	. . . . . . . . . . . . 13 |- ran F = ran S
28	12, 27	eqtri 1908	. . . . . . . . . . . 12 |- Y = ran S
29	28	grpfo 9323	. . . . . . . . . . 11 |- (S e. Grp -> S:(Y X. Y)-onto->Y)
30		fofn 4619	. . . . . . . . . . 11 |- (S:(Y X. Y)-onto->Y -> S Fn (Y X. Y))
31	29, 30	syl 12	. . . . . . . . . 10 |- (S e. Grp -> S Fn (Y X. Y))
32		fnresdm 4522	. . . . . . . . . 10 |- (S Fn (Y X. Y) -> (S |` (Y X. Y)) = S)
33	31, 32	syl 12	. . . . . . . . 9 |- (S e. Grp -> (S |` (Y X. Y)) = S)
34	18, 33	ax-mp 7	. . . . . . . 8 |- (S |` (Y X. Y)) = S
35	34	eqcomi 1888	. . . . . . 7 |- S = (S |` (Y X. Y))
36		eqid 1884	. . . . . . 7 |- (Id` S) = (Id` S)
37	3, 12, 35, 28, 10, 36	ghomf1o 13638	. . . . . 6 |- ((G e. Grp /\ S e. Grp /\ F e. (G GrpHom S)) -> (F:X-1-1-onto->Y <-> A.y e. X ((F` y) = (Id` S) -> y = U)))
38	1, 18, 22, 37	mp3an 1191	. . . . 5 |- (F:X-1-1-onto->Y <-> A.y e. X ((F` y) = (Id` S) -> y = U))
39	3, 10	grpidcl 9343	. . . . . . . . . 10 |- (G e. Grp -> U e. X)
40	1, 39	ax-mp 7	. . . . . . . . 9 |- U e. X
41	11, 3	grplactval 9405	. . . . . . . . 9 |- ((G e. Grp /\ y e. X /\ U e. X) -> ((F` y)` U) = (yGU))
42	1, 40, 41	mp3an13 1182	. . . . . . . 8 |- (y e. X -> ((F` y)` U) = (yGU))
43	3, 10	grprid 9346	. . . . . . . . 9 |- ((G e. Grp /\ y e. X) -> (yGU) = y)
44	1, 43	mpan 759	. . . . . . . 8 |- (y e. X -> (yGU) = y)
45	42, 44	eqtrd 1925	. . . . . . 7 |- (y e. X -> ((F` y)` U) = y)
46	45	eqeq1d 1892	. . . . . 6 |- (y e. X -> (((F` y)` U) = U <-> y = U))
47		eqid 1884	. . . . . . . . . . 11 |- (Id` H) = (Id` H)
48	47, 36	subgid 9429	. . . . . . . . . 10 |- (S e. (SubGrp` H) -> (Id` S) = (Id` H))
49	15, 48	ax-mp 7	. . . . . . . . 9 |- (Id` S) = (Id` H)
50	2	fveq2i 4684	. . . . . . . . . 10 |- (Id` H) = (Id` (SymGrp` X))
51	6	symgidi 10206	. . . . . . . . . 10 |- (Id` (SymGrp` X)) = ( _I |` X)
52	50, 51	eqtri 1908	. . . . . . . . 9 |- (Id` H) = ( _I |` X)
53	49, 52	eqtr2i 1909	. . . . . . . 8 |- ( _I |` X) = (Id` S)
54	53	eqeq2i 1894	. . . . . . 7 |- ((F` y) = ( _I |` X) <-> (F` y) = (Id` S))
55		fveq1 4680	. . . . . . . 8 |- ((F` y) = ( _I |` X) -> ((F` y)` U) = (( _I |` X)` U))
56		fvresi 4819	. . . . . . . . 9 |- (U e. X -> (( _I |` X)` U) = U)
57	40, 56	ax-mp 7	. . . . . . . 8 |- (( _I |` X)` U) = U
58	55, 57	syl6eq 1944	. . . . . . 7 |- ((F` y) = ( _I |` X) -> ((F` y)` U) = U)
59	54, 58	sylbir 218	. . . . . 6 |- ((F` y) = (Id` S) -> ((F` y)` U) = U)
60	46, 59	syl5bi 225	. . . . 5 |- (y e. X -> ((F` y) = (Id` S) -> y = U))
61	38, 60	mprgbir 2163	. . . 4 |- F:X-1-1-onto->Y
62		f1oeq3 4632	. . . . . 6 |- (Y = ran S -> (F:X-1-1-onto->Y <-> F:X-1-1-onto->ran S))
63	28, 62	ax-mp 7	. . . . 5 |- (F:X-1-1-onto->Y <-> F:X-1-1-onto->ran S)
64		f1oeq2 4631	. . . . . 6 |- (X = ran G -> (F:X-1-1-onto->ran S <-> F:ran G-1-1-onto->ran S))
65	3, 64	ax-mp 7	. . . . 5 |- (F:X-1-1-onto->ran S <-> F:ran G-1-1-onto->ran S)
66	63, 65	bitri 190	. . . 4 |- (F:X-1-1-onto->Y <-> F:ran G-1-1-onto->ran S)
67	61, 66	mpbi 206	. . 3 |- F:ran G-1-1-onto->ran S
68	20, 22, 67	mpbir2an 800	. 2 |- F e. (G GrpIso S)
69	15, 68	pm3.2i 307	1 |- (S e. (SubGrp` H) /\ F e. (G GrpIso S))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292   C_ wss 2593  {copab 3395   _I cid 3582   X. cxp 3984  ran crn 3987   |` cres 3988   Fn wfn 3993  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  SubGrpcsubg 9423   GrpHom cghom 10189   GrpIso cgiso 10191  SymGrpcsymgrp 10198

This theorem is referenced by:  cayleyi 13644

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-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-subg 9424  df-ghom 10190  df-giso 10192  df-symgrp 10199

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