Theorem isga2 9452

Description: An equivalent form of isga 9450. (Contributed by Jeff Hankins, 10-Aug-2009.)

Hypotheses

Ref	Expression
isga2.1	|- X = ran G
isga2.2	|- Y = ran M
isga2.3	|- U = (Id` G)

Assertion

Ref	Expression
isga2	|- (M e. A -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))

Distinct variable groups:   x,y,z,G   x,M,y,z   y,X,z   x,Y

Proof of Theorem isga2

Step	Hyp	Ref	Expression
1		simpl 346	. . 3 |- ((M e. A /\ <.G, M>. e. GrpAct) -> M e. A)
2		isga2.1	. . . . 5 |- X = ran G
3		eqid 1884	. . . . 5 |- ran dom M = ran dom M
4	2, 3	gafo 9451	. . . 4 |- ((M e. A /\ <.G, M>. e. GrpAct) -> M:(X X. ran dom M)-onto->ran dom M)
5		forn 4620	. . . . 5 |- (M:(X X. ran dom M)-onto->ran dom M -> ran M = ran dom M)
6		isga2.2	. . . . 5 |- Y = ran M
7	5, 6	syl5eq 1940	. . . 4 |- (M:(X X. ran dom M)-onto->ran dom M -> Y = ran dom M)
8	4, 7	syl 12	. . 3 |- ((M e. A /\ <.G, M>. e. GrpAct) -> Y = ran dom M)
9	1, 8	jca 310	. 2 |- ((M e. A /\ <.G, M>. e. GrpAct) -> (M e. A /\ Y = ran dom M))
10		fdm 4567	. . . . . . 7 |- (M:(X X. Y)-->Y -> dom M = (X X. Y))
11	10	rneqd 4188	. . . . . 6 |- (M:(X X. Y)-->Y -> ran dom M = ran ( X X. Y))
12	11	adantl 424	. . . . 5 |- ((G e. Grp /\ M:(X X. Y)-->Y) -> ran dom M = ran ( X X. Y))
13	2	grpn0 9326	. . . . . . 7 |- (G e. Grp -> X =/= (/))
14		rnxp 4342	. . . . . . 7 |- (X =/= (/) -> ran ( X X. Y) = Y)
15	13, 14	syl 12	. . . . . 6 |- (G e. Grp -> ran ( X X. Y) = Y)
16	15	adantr 425	. . . . 5 |- ((G e. Grp /\ M:(X X. Y)-->Y) -> ran ( X X. Y) = Y)
17	12, 16	eqtr2d 1926	. . . 4 |- ((G e. Grp /\ M:(X X. Y)-->Y) -> Y = ran dom M)
18	17	3adant3 896	. . 3 |- ((G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) -> Y = ran dom M)
19	18	anim2i 362	. 2 |- ((M e. A /\ (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))) -> (M e. A /\ Y = ran dom M))
20		isga2.3	. . . 4 |- U = (Id` G)
21	2, 3, 20	isga 9450	. . 3 |- (M e. A -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ M:(X X. ran dom M)-->ran dom M /\ A.x e. ran dom M((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
22		xpeq2 4017	. . . . . 6 |- (ran dom M = Y -> (X X. ran dom M) = (X X. Y))
23		feq23 4554	. . . . . 6 |- (((X X. ran dom M) = (X X. Y) /\ ran dom M = Y) -> (M:(X X. ran dom M)-->ran dom M <-> M:(X X. Y)-->Y))
24	22, 23	mpancom 769	. . . . 5 |- (ran dom M = Y -> (M:(X X. ran dom M)-->ran dom M <-> M:(X X. Y)-->Y))
25		raleq 2266	. . . . 5 |- (ran dom M = Y -> (A.x e. ran dom M((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))) <-> A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))))
26	24, 25	3anbi23d 1171	. . . 4 |- (ran dom M = Y -> ((G e. Grp /\ M:(X X. ran dom M)-->ran dom M /\ A.x e. ran dom M((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
27	26	eqcoms 1887	. . 3 |- (Y = ran dom M -> ((G e. Grp /\ M:(X X. ran dom M)-->ran dom M /\ A.x e. ran dom M((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx)))) <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
28	21, 27	sylan9bb 599	. 2 |- ((M e. A /\ Y = ran dom M) -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))
29	9, 19, 28	pm5.21nd 744	1 |- (M e. A -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ M:(X X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. X A.z e. X ((yGz)Mx) = (yM(zMx))))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  (/)c0 2875  <.cop 3046   X. cxp 3984  dom cdm 3986  ran crn 3987  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  GrpActcga 9447

This theorem is referenced by:  ga0 9453  gaid 9454  ssga 9455  gagrp 9456  gaf 9457  gagrpid 9458  gaass 9459  curgrpact 14735

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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