Theorem exidres 16031

Description: The restriction of a binary operation with identity to a subset containing the identity has an 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
exidres	|- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> H e. ExId )

Proof of Theorem exidres

Step	Hyp	Ref	Expression
1		exidres.1	. . . 4 |- X = ran G
2		exidres.2	. . . 4 |- U = (Id` G)
3		exidres.3	. . . 4 |- H = (G |` (Y X. Y))
4	1, 2, 3	exidreslem 16030	. . 3 |- ((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)))
5		opreq2 4890	. . . . . . 7 |- (u = U -> (xHu) = (xHU))
6	5	eqeq1d 1892	. . . . . 6 |- (u = U -> ((xHu) = x <-> (xHU) = x))
7		opreq1 4889	. . . . . . 7 |- (u = U -> (uHx) = (UHx))
8	7	eqeq1d 1892	. . . . . 6 |- (u = U -> ((uHx) = x <-> (UHx) = x))
9	6, 8	anbi12d 690	. . . . 5 |- (u = U -> (((xHu) = x /\ (uHx) = x) <-> ((xHU) = x /\ (UHx) = x)))
10	9	ralbidv 2123	. . . 4 |- (u = U -> (A.x e. dom dom H((xHu) = x /\ (uHx) = x) <-> A.x e. dom dom H((xHU) = x /\ (UHx) = x)))
11	10	rcla4ev 2381	. . 3 |- ((U e. dom dom H /\ A.x e. dom dom H((xHU) = x /\ (UHx) = x)) -> E.u e. dom dom HA.x e. dom dom H((xHu) = x /\ (uHx) = x))
12	4, 11	syl 12	. 2 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> E.u e. dom dom HA.x e. dom dom H((xHu) = x /\ (uHx) = x))
13		resexg 4250	. . . . 5 |- (G e. (Magma i^i ExId ) -> (G |` (Y X. Y)) e. _V)
14	13, 3	syl5eqel 1975	. . . 4 |- (G e. (Magma i^i ExId ) -> H e. _V)
15		eqid 1884	. . . . 5 |- dom dom H = dom dom H
16	15	isexid 10364	. . . 4 |- (H e. _V -> (H e. ExId <-> E.u e. dom dom HA.x e. dom dom H((xHu) = x /\ (uHx) = x)))
17	14, 16	syl 12	. . 3 |- (G e. (Magma i^i ExId ) -> (H e. ExId <-> E.u e. dom dom HA.x e. dom dom H((xHu) = x /\ (uHx) = x)))
18	17	3ad2ant1 897	. 2 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> (H e. ExId <-> E.u e. dom dom HA.x e. dom dom H((xHu) = x /\ (uHx) = x)))
19	12, 18	mpbird 213	1 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> H e. ExId )

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593   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:  exidresid 16032

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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