Theorem gagrpid 9458

Description: The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.)

Hypotheses

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

Assertion

Ref	Expression
gagrpid	|- ((M e. B /\ <.G, M>. e. GrpAct /\ A e. Y) -> (UMA) = A)

Proof of Theorem gagrpid

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . . 6 |- (x = A -> (UMx) = (UMA))
2		id 73	. . . . . 6 |- (x = A -> x = A)
3	1, 2	eqeq12d 1899	. . . . 5 |- (x = A -> ((UMx) = x <-> (UMA) = A))
4	3	imbi2d 674	. . . 4 |- (x = A -> (((M e. B /\ <.G, M>. e. GrpAct) -> (UMx) = x) <-> ((M e. B /\ <.G, M>. e. GrpAct) -> (UMA) = A)))
5		eqid 1884	. . . . . . . . 9 |- ran G = ran G
6		gagrpid.1	. . . . . . . . 9 |- Y = ran M
7		gagrpid.2	. . . . . . . . 9 |- U = (Id` G)
8	5, 6, 7	isga2 9452	. . . . . . . 8 |- (M e. B -> (<.G, M>. e. GrpAct <-> (G e. Grp /\ M:(ran G X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. ran GA.z e. ran G((yGz)Mx) = (yM(zMx))))))
9	8	biimpa 460	. . . . . . 7 |- ((M e. B /\ <.G, M>. e. GrpAct) -> (G e. Grp /\ M:(ran G X. Y)-->Y /\ A.x e. Y ((UMx) = x /\ A.y e. ran GA.z e. ran G((yGz)Mx) = (yM(zMx)))))
10	9	simp3d 890	. . . . . 6 |- ((M e. B /\ <.G, M>. e. GrpAct) -> A.x e. Y ((UMx) = x /\ A.y e. ran GA.z e. ran G((yGz)Mx) = (yM(zMx))))
11		ra4 2155	. . . . . 6 |- (A.x e. Y ((UMx) = x /\ A.y e. ran GA.z e. ran G((yGz)Mx) = (yM(zMx))) -> (x e. Y -> ((UMx) = x /\ A.y e. ran GA.z e. ran G((yGz)Mx) = (yM(zMx)))))
12		simpl 346	. . . . . . 7 |- (((UMx) = x /\ A.y e. ran GA.z e. ran G((yGz)Mx) = (yM(zMx))) -> (UMx) = x)
13	12	imim2i 11	. . . . . 6 |- ((x e. Y -> ((UMx) = x /\ A.y e. ran GA.z e. ran G((yGz)Mx) = (yM(zMx)))) -> (x e. Y -> (UMx) = x))
14	10, 11, 13	3syl 24	. . . . 5 |- ((M e. B /\ <.G, M>. e. GrpAct) -> (x e. Y -> (UMx) = x))
15	14	com12 14	. . . 4 |- (x e. Y -> ((M e. B /\ <.G, M>. e. GrpAct) -> (UMx) = x))
16	4, 15	vtoclga 2352	. . 3 |- (A e. Y -> ((M e. B /\ <.G, M>. e. GrpAct) -> (UMA) = A))
17	16	com12 14	. 2 |- ((M e. B /\ <.G, M>. e. GrpAct) -> (A e. Y -> (UMA) = A))
18	17	3impia 1064	1 |- ((M e. B /\ <.G, M>. e. GrpAct /\ A e. Y) -> (UMA) = A)

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

This theorem is referenced by:  gacan 9460

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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