Theorem gafo 9451

Description: A group action is onto its base set. (Contributed by Jeff Hankins, 10-Aug-2009.)

Hypotheses

Ref	Expression
gafo.1	|- X = ran G
gafo.2	|- Y = ran dom M

Assertion

Ref	Expression
gafo	|- ((M e. A /\ <.G, M>. e. GrpAct) -> M:(X X. Y)-onto->Y)

Proof of Theorem gafo

Step	Hyp	Ref	Expression
1		gafo.1	. . . . 5 |- X = ran G
2		gafo.2	. . . . 5 |- Y = ran dom M
3		eqid 1884	. . . . 5 |- (Id` G) = (Id` G)
4	1, 2, 3	isga 9450	. . . 4 |- (M e. A -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
5	4	biimpa 460	. . 3 |- ((M e. A /\ <.G, M>. e. GrpAct) -> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))
6		ffn 4562	. . . . 5 |- (M:(X X. Y)-->Y -> M Fn (X X. Y))
7	6	3ad2ant2 898	. . . 4 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> M Fn (X X. Y))
8		frn 4569	. . . . . 6 |- (M:(X X. Y)-->Y -> ran M C_ Y)
9	8	3ad2ant2 898	. . . . 5 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> ran M C_ Y)
10		ax-17 1317	. . . . . . . 8 |- (G e. Grp -> A.x G e. Grp)
11		ax-17 1317	. . . . . . . 8 |- (M:(X X. Y)-->Y -> A.x M:(X X. Y)-->Y)
12		hbra1 2147	. . . . . . . 8 |- (A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) -> A.xA.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))
13	10, 11, 12	hb3an 1359	. . . . . . 7 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> A.x(G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))
14		ra4 2155	. . . . . . . . . 10 |- (A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) -> (x e. Y -> (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))
15		simpl2l 929	. . . . . . . . . . . 12 |- (((x e. Y /\ (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) /\ G e. Grp) /\ M:(X X. Y)-->Y) -> ((Id` G)Mx) = x)
16	6	adantl 424	. . . . . . . . . . . . 13 |- (((x e. Y /\ (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) /\ G e. Grp) /\ M:(X X. Y)-->Y) -> M Fn (X X. Y))
17	1, 3	grpidcl 9343	. . . . . . . . . . . . . . 15 |- (G e. Grp -> (Id` G) e. X)
18	17	3ad2ant3 899	. . . . . . . . . . . . . 14 |- ((x e. Y /\ (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) /\ G e. Grp) -> (Id` G) e. X)
19	18	adantr 425	. . . . . . . . . . . . 13 |- (((x e. Y /\ (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) /\ G e. Grp) /\ M:(X X. Y)-->Y) -> (Id` G) e. X)
20		simpl1 879	. . . . . . . . . . . . 13 |- (((x e. Y /\ (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) /\ G e. Grp) /\ M:(X X. Y)-->Y) -> x e. Y)
21		fnoprvrn 4968	. . . . . . . . . . . . 13 |- ((M Fn (X X. Y) /\ (Id` G) e. X /\ x e. Y) -> ((Id` G)Mx) e. ran M)
22	16, 19, 20, 21	syl111anc 1100	. . . . . . . . . . . 12 |- (((x e. Y /\ (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) /\ G e. Grp) /\ M:(X X. Y)-->Y) -> ((Id` G)Mx) e. ran M)
23	15, 22	eqeltrrd 1972	. . . . . . . . . . 11 |- (((x e. Y /\ (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) /\ G e. Grp) /\ M:(X X. Y)-->Y) -> x e. ran M)
24	23	3exp1 1084	. . . . . . . . . 10 |- (x e. Y -> ((((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) -> (G e. Grp -> (M:(X X. Y)-->Y -> x e. ran M))))
25	14, 24	sylcom 62	. . . . . . . . 9 |- (A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) -> (x e. Y -> (G e. Grp -> (M:(X X. Y)-->Y -> x e. ran M))))
26	25	com4t 44	. . . . . . . 8 |- (G e. Grp -> (M:(X X. Y)-->Y -> (A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) -> (x e. Y -> x e. ran M))))
27	26	3imp 1061	. . . . . . 7 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> (x e. Y -> x e. ran M))
28	13, 27	19.21ai 1345	. . . . . 6 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> A.x(x e. Y -> x e. ran M))
29		dfss2 2610	. . . . . 6 |- (Y C_ ran M <-> A.x(x e. Y -> x e. ran M))
30	28, 29	sylibr 217	. . . . 5 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> Y C_ ran M)
31	9, 30	eqssd 2633	. . . 4 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> ran M = Y)
32	7, 31	jca 310	. . 3 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y (((Id` G)Mx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> (M Fn (X X. Y) /\ ran M = Y))
33	5, 32	syl 12	. 2 |- ((M e. A /\ <.G, M>. e. GrpAct) -> (M Fn (X X. Y) /\ ran M = Y))
34		df-fo 4012	. 2 |- (M:(X X. Y)-onto->Y <-> (M Fn (X X. Y) /\ ran M = Y))
35	33, 34	sylibr 217	1 |- ((M e. A /\ <.G, M>. e. GrpAct) -> M:(X X. Y)-onto->Y)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105   C_ wss 2593  <.cop 3046   X. cxp 3984  dom cdm 3986  ran crn 3987   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  GrpActcga 9447

This theorem is referenced by:  isga2 9452

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-ga 9448

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