Theorem grpidval 9342

Description: The value of the identity element of a group.

Hypotheses

Ref	Expression
grpidval.1	|- X = ran G
grpidval.2	|- U = (Id` G)

Assertion

Ref	Expression
grpidval	|- (G e. Grp -> U = U.{u e. X | A.x e. X (uGx) = x})

Distinct variable groups:   x,u,G   u,U,x   u,X,x

Proof of Theorem grpidval

Step	Hyp	Ref	Expression
1		fveq2 4681	. . . 4 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> (Id` G) = (Id` if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})))
2		rneq 4186	. . . . . . 7 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> ran G = ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}))
3		rabeq 2289	. . . . . . 7 |- (ran G = ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> {u e. ran G | A.x e. ran G(uGx) = x} = {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran G(uGx) = x})
4	2, 3	syl 12	. . . . . 6 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> {u e. ran G | A.x e. ran G(uGx) = x} = {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran G(uGx) = x})
5		opreq 4888	. . . . . . . . 9 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> (uGx) = (uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x))
6	5	eqeq1d 1892	. . . . . . . 8 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> ((uGx) = x <-> (uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x))
7	2, 6	raleqbidv 2274	. . . . . . 7 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> (A.x e. ran G(uGx) = x <-> A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x))
8	7	rabbidv 2287	. . . . . 6 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran G(uGx) = x} = {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x})
9	4, 8	eqtrd 1925	. . . . 5 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> {u e. ran G | A.x e. ran G(uGx) = x} = {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x})
10	9	unieqd 3188	. . . 4 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> U.{u e. ran G | A.x e. ran G(uGx) = x} = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x})
11	1, 10	eqeq12d 1899	. . 3 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> ((Id` G) = U.{u e. ran G | A.x e. ran G(uGx) = x} <-> (Id` if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})) = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x}))
12		grpidval.2	. . . 4 |- U = (Id` G)
13		grpidval.1	. . . . . . 7 |- X = ran G
14		rabeq 2289	. . . . . . 7 |- (X = ran G -> {u e. X | A.x e. X (uGx) = x} = {u e. ran G | A.x e. X (uGx) = x})
15	13, 14	ax-mp 7	. . . . . 6 |- {u e. X | A.x e. X (uGx) = x} = {u e. ran G | A.x e. X (uGx) = x}
16		raleq 2266	. . . . . . . . 9 |- (X = ran G -> (A.x e. X (uGx) = x <-> A.x e. ran G(uGx) = x))
17	13, 16	ax-mp 7	. . . . . . . 8 |- (A.x e. X (uGx) = x <-> A.x e. ran G(uGx) = x)
18	17	a1i 8	. . . . . . 7 |- (u e. ran G -> (A.x e. X (uGx) = x <-> A.x e. ran G(uGx) = x))
19	18	rabbiia 2285	. . . . . 6 |- {u e. ran G | A.x e. X (uGx) = x} = {u e. ran G | A.x e. ran G(uGx) = x}
20	15, 19	eqtri 1908	. . . . 5 |- {u e. X | A.x e. X (uGx) = x} = {u e. ran G | A.x e. ran G(uGx) = x}
21	20	unieqi 3187	. . . 4 |- U.{u e. X | A.x e. X (uGx) = x} = U.{u e. ran G | A.x e. ran G(uGx) = x}
22	12, 21	eqeq12i 1897	. . 3 |- (U = U.{u e. X | A.x e. X (uGx) = x} <-> (Id` G) = U.{u e. ran G | A.x e. ran G(uGx) = x})
23	11, 22	syl5bb 591	. 2 |- (G = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> (U = U.{u e. X | A.x e. X (uGx) = x} <-> (Id` if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})) = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x}))
24		df-gid 9317	. . . 4 |- Id = {<.g, y>. | y = U.{u e. ran g | A.x e. ran g((ugx) = x /\ (xgu) = x)}}
25	24	fveq1i 4682	. . 3 |- (Id` if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})) = ({<.g, y>. | y = U.{u e. ran g | A.x e. ran g((ugx) = x /\ (xgu) = x)}}` if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}))
26		elisset 2299	. . . . . . . . 9 |- (G e. Grp -> G e. _V)
27	26	ancli 320	. . . . . . . 8 |- (G e. Grp -> (G e. Grp /\ G e. _V))
28	27	con3i 114	. . . . . . 7 |- (-. (G e. Grp /\ G e. _V) -> -. G e. Grp)
29		snex 3492	. . . . . . 7 |- {<.<.(/), (/)>., (/)>.} e. _V
30	28, 29	jctir 317	. . . . . 6 |- (-. (G e. Grp /\ G e. _V) -> (-. G e. Grp /\ {<.<.(/), (/)>., (/)>.} e. _V))
31	30	orri 248	. . . . 5 |- ((G e. Grp /\ G e. _V) \/ (-. G e. Grp /\ {<.<.(/), (/)>., (/)>.} e. _V))
32		ifel 3006	. . . . 5 |- (if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) e. _V <-> ((G e. Grp /\ G e. _V) \/ (-. G e. Grp /\ {<.<.(/), (/)>., (/)>.} e. _V)))
33	31, 32	mpbir 207	. . . 4 |- if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) e. _V
34	33	rnex 4209	. . . . . 6 |- ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) e. _V
35	34	rabex 3461	. . . . 5 |- {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x} e. _V
36	35	uniex 3794	. . . 4 |- U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x} e. _V
37		rneq 4186	. . . . . . . 8 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> ran g = ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}))
38		rabeq 2289	. . . . . . . 8 |- (ran g = ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> {u e. ran g | A.x e. ran g((ugx) = x /\ (xgu) = x)} = {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran g((ugx) = x /\ (xgu) = x)})
39	37, 38	syl 12	. . . . . . 7 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> {u e. ran g | A.x e. ran g((ugx) = x /\ (xgu) = x)} = {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran g((ugx) = x /\ (xgu) = x)})
40		opreq 4888	. . . . . . . . . . 11 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> (ugx) = (uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x))
41	40	eqeq1d 1892	. . . . . . . . . 10 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> ((ugx) = x <-> (uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x))
42		opreq 4888	. . . . . . . . . . 11 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> (xgu) = (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u))
43	42	eqeq1d 1892	. . . . . . . . . 10 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> ((xgu) = x <-> (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u) = x))
44	41, 43	anbi12d 690	. . . . . . . . 9 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> (((ugx) = x /\ (xgu) = x) <-> ((uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x /\ (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u) = x)))
45	37, 44	raleqbidv 2274	. . . . . . . 8 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> (A.x e. ran g((ugx) = x /\ (xgu) = x) <-> A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})((uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x /\ (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u) = x)))
46	45	rabbidv 2287	. . . . . . 7 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran g((ugx) = x /\ (xgu) = x)} = {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})((uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x /\ (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u) = x)})
47	39, 46	eqtrd 1925	. . . . . 6 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> {u e. ran g | A.x e. ran g((ugx) = x /\ (xgu) = x)} = {u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})((uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x /\ (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u) = x)})
48	47	unieqd 3188	. . . . 5 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> U.{u e. ran g | A.x e. ran g((ugx) = x /\ (xgu) = x)} = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})((uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x /\ (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u) = x)})
49		0ex 3446	. . . . . . . 8 |- (/) e. _V
50	49	grpsn 9340	. . . . . . 7 |- {<.<.(/), (/)>., (/)>.} e. Grp
51	50	elimel 3025	. . . . . 6 |- if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) e. Grp
52		eqid 1884	. . . . . . 7 |- ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) = ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})
53	52	grprlidrid 9337	. . . . . 6 |- (if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) e. Grp -> U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})((uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x /\ (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u) = x)} = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x})
54	51, 53	ax-mp 7	. . . . 5 |- U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})((uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x /\ (xif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})u) = x)} = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x}
55	48, 54	syl6eq 1944	. . . 4 |- (g = if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) -> U.{u e. ran g | A.x e. ran g((ugx) = x /\ (xgu) = x)} = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x})
56	33, 36, 55	fvopab 4753	. . 3 |- ({<.g, y>. | y = U.{u e. ran g | A.x e. ran g((ugx) = x /\ (xgu) = x)}}` if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})) = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x}
57	25, 56	eqtri 1908	. 2 |- (Id` if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})) = U.{u e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.}) | A.x e. ran if(G e. Grp, G, {<.<.(/), (/)>., (/)>.})(uif(G e. Grp, G, {<.<.(/), (/)>., (/)>.})x) = x}
58	23, 57	dedth 3011	1 |- (G e. Grp -> U = U.{u e. X | A.x e. X (uGx) = x})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292  (/)c0 2875  ifcif 2982  {csn 3044  <.cop 3046  U.cuni 3177  {copab 3395  ran crn 3987  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312

This theorem is referenced by:  grpidcl 9343  grpidinv2 9344  cnid 9435  mulid 9440  zrdivrng 10418  hilid 10661

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

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