Theorem grpidval 8054

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 3365	. . . . 5 |- (G e. Grp -> ran G e. V)
2		grpidval.1	. . . . 5 |- X = ran G
3	1, 2	syl5eqel 1555	. . . 4 |- (G e. Grp -> X e. V)
4		rabexg 2729	. . . 4 |- (X e. V -> {u e. X | A.x e. X (uGx) = x} e. V)
5		uniexg 2877	. . . 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 3345	. . . . . . . 8 |- (g = G -> ran g = ran G)
8	7, 2	syl6eqr 1528	. . . . . . 7 |- (g = G -> ran g = X)
9		rabeq 1812	. . . . . . 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 3973	. . . . . . . . 9 |- (g = G -> (ugx) = (uGx))
12	11	eqeq1d 1486	. . . . . . . 8 |- (g = G -> ((ugx) = x <-> (uGx) = x))
13	8, 12	raleq12d 1797	. . . . . . 7 |- (g = G -> (A.x e. ran g(ugx) = x <-> A.x e. X (uGx) = x))
14	13	rabbisdv 1810	. . . . . 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 1510	. . . . 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 2516	. . . 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 8035	. . . 4 |- Id = {<.g, y>. | (g e. Grp /\ y = U.{u e. ran g | A.x e. ran g(ugx) = x})}
18	16, 17	fvopab4g 3785	. . 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 706	. 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 1522	1 |- (G e. Grp -> U = U.{u e. X | A.x e. X (uGx) = x})

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  A.wral 1648  {crab 1651  Vcvv 1814  U.cuni 2507  ran crn 3177  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031

This theorem is referenced by:  grpidcl 8055  grpidinv2 8056  cnid 8123  mulid 8128  hilid 9023

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opr 3971  df-gid 8035

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