Theorem exidresid 16032

Description: The restriction of a binary operation with identity to a subset containing the identity has the same identity element.

Hypotheses

Ref	Expression
exidres.1	|- X = ran G
exidres.2	|- U = (Id` G)
exidres.3	|- H = (G |` (Y X. Y))

Assertion

Ref	Expression
exidresid	|- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> (Id` H) = U)

Proof of Theorem exidresid

Step	Hyp	Ref	Expression
1		resexg 4250	. . . . . 6 |- (G e. (Magma i^i ExId ) -> (G |` (Y X. Y)) e. _V)
2		exidres.3	. . . . . 6 |- H = (G |` (Y X. Y))
3	1, 2	syl5eqel 1975	. . . . 5 |- (G e. (Magma i^i ExId ) -> H e. _V)
4		eqid 1884	. . . . . 6 |- ran H = ran H
5		eqid 1884	. . . . . 6 |- (Id` H) = (Id` H)
6	4, 5	idrval 10374	. . . . 5 |- (H e. _V -> (Id` H) = U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
7	3, 6	syl 12	. . . 4 |- (G e. (Magma i^i ExId ) -> (Id` H) = U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
8	7	3ad2ant1 897	. . 3 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> (Id` H) = U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
9	8	adantr 425	. 2 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> (Id` H) = U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)})
10		exidres.1	. . . . . . 7 |- X = ran G
11		exidres.2	. . . . . . 7 |- U = (Id` G)
12	10, 11, 2	exidreslem 16030	. . . . . 6 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> (U e. dom dom H /\ A.x e. dom dom H((xHU) = x /\ (UHx) = x)))
13	12	simprd 352	. . . . 5 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> A.x e. dom dom H((xHU) = x /\ (UHx) = x))
14	13	adantr 425	. . . 4 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> A.x e. dom dom H((xHU) = x /\ (UHx) = x))
15		elin 2786	. . . . . . . 8 |- (H e. (Magma i^i ExId ) <-> (H e. Magma /\ H e. ExId ))
16		rngopid 10370	. . . . . . . 8 |- (H e. (Magma i^i ExId ) -> ran H = dom dom H)
17	15, 16	sylbir 218	. . . . . . 7 |- ((H e. Magma /\ H e. ExId ) -> ran H = dom dom H)
18	17	ancoms 484	. . . . . 6 |- ((H e. ExId /\ H e. Magma) -> ran H = dom dom H)
19	10, 11, 2	exidres 16031	. . . . . 6 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> H e. ExId )
20	18, 19	sylan 497	. . . . 5 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> ran H = dom dom H)
21		ancom 482	. . . . . 6 |- (((UHx) = x /\ (xHU) = x) <-> ((xHU) = x /\ (UHx) = x))
22	21	a1i 8	. . . . 5 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> (((UHx) = x /\ (xHU) = x) <-> ((xHU) = x /\ (UHx) = x)))
23	20, 22	raleqbidv 2274	. . . 4 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> (A.x e. ran H((UHx) = x /\ (xHU) = x) <-> A.x e. dom dom H((xHU) = x /\ (UHx) = x)))
24	14, 23	mpbird 213	. . 3 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> A.x e. ran H((UHx) = x /\ (xHU) = x))
25	12	simplld 348	. . . . . 6 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> U e. dom dom H)
26	25	adantr 425	. . . . 5 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> U e. dom dom H)
27	26, 20	eleqtrrd 1974	. . . 4 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> U e. ran H)
28	4	exidu1 10373	. . . . . . 7 |- (H e. (Magma i^i ExId ) -> E!u e. ran HA.x e. ran H((uHx) = x /\ (xHu) = x))
29	15, 28	sylbir 218	. . . . . 6 |- ((H e. Magma /\ H e. ExId ) -> E!u e. ran HA.x e. ran H((uHx) = x /\ (xHu) = x))
30	29	ancoms 484	. . . . 5 |- ((H e. ExId /\ H e. Magma) -> E!u e. ran HA.x e. ran H((uHx) = x /\ (xHu) = x))
31	30, 19	sylan 497	. . . 4 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> E!u e. ran HA.x e. ran H((uHx) = x /\ (xHu) = x))
32		opreq1 4889	. . . . . . . 8 |- (u = U -> (uHx) = (UHx))
33	32	eqeq1d 1892	. . . . . . 7 |- (u = U -> ((uHx) = x <-> (UHx) = x))
34		opreq2 4890	. . . . . . . 8 |- (u = U -> (xHu) = (xHU))
35	34	eqeq1d 1892	. . . . . . 7 |- (u = U -> ((xHu) = x <-> (xHU) = x))
36	33, 35	anbi12d 690	. . . . . 6 |- (u = U -> (((uHx) = x /\ (xHu) = x) <-> ((UHx) = x /\ (xHU) = x)))
37	36	ralbidv 2123	. . . . 5 |- (u = U -> (A.x e. ran H((uHx) = x /\ (xHu) = x) <-> A.x e. ran H((UHx) = x /\ (xHU) = x)))
38	37	reuuni2 3811	. . . 4 |- ((U e. ran H /\ E!u e. ran HA.x e. ran H((uHx) = x /\ (xHu) = x)) -> (A.x e. ran H((UHx) = x /\ (xHU) = x) <-> U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)} = U))
39	27, 31, 38	syl11anc 524	. . 3 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> (A.x e. ran H((UHx) = x /\ (xHU) = x) <-> U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)} = U))
40	24, 39	mpbid 212	. 2 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> U.{u e. ran H | A.x e. ran H((uHx) = x /\ (xHu) = x)} = U)
41	9, 40	eqtrd 1925	1 |- (((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) /\ H e. Magma) -> (Id` H) = U)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E!wreu 2107  {crab 2108  _Vcvv 2292   i^i cin 2592   C_ wss 2593  U.cuni 3177   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988  ` cfv 3998  (class class class)co 4884  Idcgi 9312   ExId cexid 10361  Magmacmagm 10365

This theorem is referenced by:  isdivrng2 16111

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-reu 2111  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-gid 9317  df-exid 10362  df-mgm 10366

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