Theorem rngunval2 14774

Description: The value of the unit of a ring.

Hypotheses

Ref	Expression
rngunval2.1	|- X = ran (1st` R)
rngunval2.2	|- H = (2nd` R)
rngunval2.3	|- U = (Id` H)

Assertion

Ref	Expression
rngunval2	|- (R e. Ring -> U = U.{u e. X | A.x e. X ((uHx) = x /\ (xHu) = x)})

Distinct variable groups:   u,H,x   u,X,x

Proof of Theorem rngunval2

Step	Hyp	Ref	Expression
1		rngunval2.2	. . . . 5 |- H = (2nd` R)
2	1	a1i 8	. . . 4 |- (R e. Ring -> H = (2nd` R))
3		fvex 4689	. . . 4 |- (2nd` R) e. _V
4	2, 3	syl6eqel 1979	. . 3 |- (R e. Ring -> H e. _V)
5		eqid 1884	. . . 4 |- ran H = ran H
6		rngunval2.3	. . . 4 |- U = (Id` H)
7	5, 6	idrval 10374	. . 3 |- (H e. _V -> U = U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
8	4, 7	syl 12	. 2 |- (R e. Ring -> U = U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
9		eqid 1884	. . . . . 6 |- (1st` R) = (1st` R)
10	1, 9	rnplrnml 10404	. . . . 5 |- (R e. Ring -> ran (1st` R) = ran H)
11		rngunval2.1	. . . . 5 |- X = ran (1st` R)
12	10, 11	syl5eq 1940	. . . 4 |- (R e. Ring -> X = ran H)
13		rabeq 2289	. . . . 5 |- (X = ran H -> {u e. X | A.x e. X ((uHx) = x /\ (xHu) = x)} = {u e. ran H | A.x e. X ((uHx) = x /\ (xHu) = x)})
14		raleq 2266	. . . . . 6 |- (X = ran H -> (A.x e. X ((uHx) = x /\ (xHu) = x) <-> A.x e. ran H((uHx) = x /\ (xHu) = x)))
15	14	rabbidv 2287	. . . . 5 |- (X = ran H -> {u e. ran H | A.x e. X ((uHx) = x /\ (xHu) = x)} = {u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
16	13, 15	eqtrd 1925	. . . 4 |- (X = ran H -> {u e. X | A.x e. X ((uHx) = x /\ (xHu) = x)} = {u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
17	12, 16	syl 12	. . 3 |- (R e. Ring -> {u e. X | A.x e. X ((uHx) = x /\ (xHu) = x)} = {u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
18	17	unieqd 3188	. 2 |- (R e. Ring -> U.{u e. X | A.x e. X ((uHx) = x /\ (xHu) = x)} = U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
19	8, 18	eqtr4d 1928	1 |- (R e. Ring -> U = U.{u e. X | A.x e. X ((uHx) = x /\ (xHu) = x)})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292  U.cuni 3177  ran crn 3987  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Ringcring 9463

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-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-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  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-fo 4012  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-gid 9317  df-ring 9464

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