Theorem grpidval 8058

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		rnexg 3359	. . . . 5 |- (G e. Grp -> ran G e. V)
2		grpidval.1	. . . . 5 |- X = ran G
3	1, 2	syl5eqel 1552	. . . 4 |- (G e. Grp -> X e. V)
4		rabexg 2724	. . . 4 |- (X e. V -> {u e. X | A.x e. X (uGx) = x} e. V)
5		uniexg 2871	. . . 4 |- ({u e. X | A.x e. X (uGx) = x} e. V -> U.{u e. X | A.x e. X (uGx) = x} e. V)
6	3, 4, 5	3syl 20	. . 3 |- (G e. Grp -> U.{u e. X | A.x e. X (uGx) = x} e. V)
7		rneq 3339	. . . . . . . 8 |- (g = G -> ran g = ran G)
8	7, 2	syl6eqr 1525	. . . . . . 7 |- (g = G -> ran g = X)
9		rabeq 1809	. . . . . . 7 |- (ran g = X -> {u e. ran g | A.x e. ran g(ugx) = x} = {u e. X | A.x e. ran g(ugx) = x})
10	8, 9	syl 10	. . . . . 6 |- (g = G -> {u e. ran g | A.x e. ran g(ugx) = x} = {u e. X | A.x e. ran g(ugx) = x})
11		opreq 3967	. . . . . . . . 9 |- (g = G -> (ugx) = (uGx))
12	11	eqeq1d 1483	. . . . . . . 8 |- (g = G -> ((ugx) = x <-> (uGx) = x))
13	8, 12	raleq12d 1794	. . . . . . 7 |- (g = G -> (A.x e. ran g(ugx) = x <-> A.x e. X (uGx) = x))
14	13	rabbisdv 1807	. . . . . 6 |- (g = G -> {u e. X | A.x e. ran g(ugx) = x} = {u e. X | A.x e. X (uGx) = x})
15	10, 14	eqtrd 1507	. . . . 5 |- (g = G -> {u e. ran g | A.x e. ran g(ugx) = x} = {u e. X | A.x e. X (uGx) = x})
16	15	unieqd 2512	. . . 4 |- (g = G -> U.{u e. ran g | A.x e. ran g(ugx) = x} = U.{u e. X | A.x e. X (uGx) = x})
17		df-gid 8038	. . . 4 |- Id = {<.g, y>. | (g e. Grp /\ y = U.{u e. ran g | A.x e. ran g(ugx) = x})}
18	16, 17	fvopab4g 3779	. . 3 |- ((G e. Grp /\ U.{u e. X | A.x e. X (uGx) = x} e. V) -> (Id` G) = U.{u e. X | A.x e. X (uGx) = x})
19	6, 18	mpdan 704	. 2 |- (G e. Grp -> (Id` G) = U.{u e. X | A.x e. X (uGx) = x})
20		grpidval.2	. 2 |- U = (Id` G)
21	19, 20	syl5eq 1519	1 |- (G e. Grp -> U = U.{u e. X | A.x e. X (uGx) = x})

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648  Vcvv 1811  U.cuni 2503  ran crn 3171  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034

This theorem is referenced by:  grpidcl 8059  grpidinv2 8060  cnid 8127  mulid 8132  hilid 9028

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opr 3965  df-gid 8038

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