Theorem cayleyi 10420

Description: Cayley's Theorem (constructive version): given group G, F is an isomorphism between G and the subgroup S of the symmetry group H on the underlying set X of G. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypotheses

Ref	Expression
cayleyi.1	|- G e. Grp
cayleyi.2	|- X = ran G
cayleyi.3	|- H = (SymGrp` X)
cayleyi.4	|- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
cayleyi.5	|- Y = ran F
cayleyi.6	|- S = (H |` (Y X. Y))

Assertion

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

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

Proof of Theorem cayleyi

Step	Hyp	Ref	Expression
1		cayleyi.1	. 2 |- G e. Grp
2		eqid 1482	. 2 |- {f | f:X-1-1-onto->X} = {f | f:X-1-1-onto->X}
3		cayleyi.2	. 2 |- X = ran G
4		eqid 1482	. 2 |- (Id` G) = (Id` G)
5		cayleyi.3	. 2 |- H = (SymGrp` X)
6		cayleyi.4	. 2 |- F = {<.g, h>. | (g e. X /\ h = {<.a, b>. | (a e. X /\ b = (gGa))})}
7		cayleyi.5	. 2 |- Y = ran F
8		cayleyi.6	. 2 |- S = (H |` (Y X. Y))
9	1, 2, 3, 4, 5, 6, 7, 8	cayleylem3 10419	1 |- (S e. (SubGrp` H) /\ F e. (G GrpIso S))

Syntax hints:   /\ wa 223   = wceq 960   e. wcel 962  {cab 1470  {copab 2679   X. cxp 3182  ran crn 3185   |` cres 3186  -1-1-onto->wf1o 3195  ` cfv 3196  (class class class)co 3977  Grpcgr 8042  Idcgi 8043  SubGrpcsubg 8122   GrpIso cgiso 10387  SymGrpcsymgrp 10407

This theorem is referenced by:  cayleythlem 10421

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-nul 2723  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-reu 1658  df-rab 1659  df-v 1819  df-sbc 1949  df-csb 2010  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-if 2372  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-id 2849  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-f 3208  df-f1 3209  df-fo 3210  df-f1o 3211  df-fv 3212  df-opr 3979  df-oprab 3980  df-1st 4093  df-2nd 4094  df-grp 8046  df-gid 8047  df-ginv 8048  df-subg 8123  df-ghom 10388  df-giso 10389  df-symgrp 10408

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