Theorem exidreslem 16030

Description: Lemma for exidres 16031 and exidresid 16032.

Hypotheses

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

Assertion

Ref	Expression
exidreslem	|- ((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)))

Distinct variable groups:   x,G   x,Y   x,X   x,U   x,H

Proof of Theorem exidreslem

Step	Hyp	Ref	Expression
1		exidres.1	. . . . . . . . . . . . 13 |- X = ran G
2	1	opidon2 10371	. . . . . . . . . . . 12 |- (G e. (Magma i^i ExId ) -> G:(X X. X)-onto->X)
3		fof 4617	. . . . . . . . . . . 12 |- (G:(X X. X)-onto->X -> G:(X X. X)-->X)
4		fdm 4567	. . . . . . . . . . . 12 |- (G:(X X. X)-->X -> dom G = (X X. X))
5	2, 3, 4	3syl 24	. . . . . . . . . . 11 |- (G e. (Magma i^i ExId ) -> dom G = (X X. X))
6	5	sseq2d 2645	. . . . . . . . . 10 |- (G e. (Magma i^i ExId ) -> ((Y X. Y) C_ dom G <-> (Y X. Y) C_ (X X. X)))
7		xpss12 4089	. . . . . . . . . . 11 |- ((Y C_ X /\ Y C_ X) -> (Y X. Y) C_ (X X. X))
8	7	anidms 480	. . . . . . . . . 10 |- (Y C_ X -> (Y X. Y) C_ (X X. X))
9	6, 8	syl5bir 227	. . . . . . . . 9 |- (G e. (Magma i^i ExId ) -> (Y C_ X -> (Y X. Y) C_ dom G))
10	9	imp 377	. . . . . . . 8 |- ((G e. (Magma i^i ExId ) /\ Y C_ X) -> (Y X. Y) C_ dom G)
11		ssdmres 4235	. . . . . . . 8 |- ((Y X. Y) C_ dom G <-> dom ( G |` (Y X. Y)) = (Y X. Y))
12	10, 11	sylib 215	. . . . . . 7 |- ((G e. (Magma i^i ExId ) /\ Y C_ X) -> dom ( G |` (Y X. Y)) = (Y X. Y))
13		exidres.3	. . . . . . . 8 |- H = (G |` (Y X. Y))
14	13	dmeqi 4158	. . . . . . 7 |- dom H = dom ( G |` (Y X. Y))
15	12, 14	syl5eq 1940	. . . . . 6 |- ((G e. (Magma i^i ExId ) /\ Y C_ X) -> dom H = (Y X. Y))
16	15	dmeqd 4159	. . . . 5 |- ((G e. (Magma i^i ExId ) /\ Y C_ X) -> dom dom H = dom ( Y X. Y))
17		dmxpid 4179	. . . . 5 |- dom ( Y X. Y) = Y
18	16, 17	syl6eq 1944	. . . 4 |- ((G e. (Magma i^i ExId ) /\ Y C_ X) -> dom dom H = Y)
19	18	eleq2d 1964	. . 3 |- ((G e. (Magma i^i ExId ) /\ Y C_ X) -> (U e. dom dom H <-> U e. Y))
20	19	biimp3ar 1195	. 2 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> U e. dom dom H)
21		exidres.2	. . . . . . . . . . . 12 |- U = (Id` G)
22	1, 21	cmpidelt 10376	. . . . . . . . . . 11 |- ((G e. (Magma i^i ExId ) /\ x e. X) -> ((UGx) = x /\ (xGU) = x))
23		ssel2 2616	. . . . . . . . . . 11 |- ((Y C_ X /\ x e. Y) -> x e. X)
24	22, 23	sylan2 500	. . . . . . . . . 10 |- ((G e. (Magma i^i ExId ) /\ (Y C_ X /\ x e. Y)) -> ((UGx) = x /\ (xGU) = x))
25	24	anassrs 489	. . . . . . . . 9 |- (((G e. (Magma i^i ExId ) /\ Y C_ X) /\ x e. Y) -> ((UGx) = x /\ (xGU) = x))
26	25	adantrl 430	. . . . . . . 8 |- (((G e. (Magma i^i ExId ) /\ Y C_ X) /\ (U e. Y /\ x e. Y)) -> ((UGx) = x /\ (xGU) = x))
27	26	ancomd 483	. . . . . . 7 |- (((G e. (Magma i^i ExId ) /\ Y C_ X) /\ (U e. Y /\ x e. Y)) -> ((xGU) = x /\ (UGx) = x))
28		oprvres 4963	. . . . . . . . . . . 12 |- ((x e. Y /\ U e. Y) -> (x(G |` (Y X. Y))U) = (xGU))
29	13	opreqi 4896	. . . . . . . . . . . 12 |- (xHU) = (x(G |` (Y X. Y))U)
30	28, 29	syl5eq 1940	. . . . . . . . . . 11 |- ((x e. Y /\ U e. Y) -> (xHU) = (xGU))
31	30	ancoms 484	. . . . . . . . . 10 |- ((U e. Y /\ x e. Y) -> (xHU) = (xGU))
32	31	eqeq1d 1892	. . . . . . . . 9 |- ((U e. Y /\ x e. Y) -> ((xHU) = x <-> (xGU) = x))
33		oprvres 4963	. . . . . . . . . . 11 |- ((U e. Y /\ x e. Y) -> (U(G |` (Y X. Y))x) = (UGx))
34	13	opreqi 4896	. . . . . . . . . . 11 |- (UHx) = (U(G |` (Y X. Y))x)
35	33, 34	syl5eq 1940	. . . . . . . . . 10 |- ((U e. Y /\ x e. Y) -> (UHx) = (UGx))
36	35	eqeq1d 1892	. . . . . . . . 9 |- ((U e. Y /\ x e. Y) -> ((UHx) = x <-> (UGx) = x))
37	32, 36	anbi12d 690	. . . . . . . 8 |- ((U e. Y /\ x e. Y) -> (((xHU) = x /\ (UHx) = x) <-> ((xGU) = x /\ (UGx) = x)))
38	37	adantl 424	. . . . . . 7 |- (((G e. (Magma i^i ExId ) /\ Y C_ X) /\ (U e. Y /\ x e. Y)) -> (((xHU) = x /\ (UHx) = x) <-> ((xGU) = x /\ (UGx) = x)))
39	27, 38	mpbird 213	. . . . . 6 |- (((G e. (Magma i^i ExId ) /\ Y C_ X) /\ (U e. Y /\ x e. Y)) -> ((xHU) = x /\ (UHx) = x))
40	39	anassrs 489	. . . . 5 |- ((((G e. (Magma i^i ExId ) /\ Y C_ X) /\ U e. Y) /\ x e. Y) -> ((xHU) = x /\ (UHx) = x))
41	40	r19.21aiva 2176	. . . 4 |- (((G e. (Magma i^i ExId ) /\ Y C_ X) /\ U e. Y) -> A.x e. Y ((xHU) = x /\ (UHx) = x))
42	41	3impa 1062	. . 3 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> A.x e. Y ((xHU) = x /\ (UHx) = x))
43	10	3adant3 896	. . . . . . . 8 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> (Y X. Y) C_ dom G)
44	43, 11	sylib 215	. . . . . . 7 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> dom ( G |` (Y X. Y)) = (Y X. Y))
45	44, 14	syl5eq 1940	. . . . . 6 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> dom H = (Y X. Y))
46	45	dmeqd 4159	. . . . 5 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> dom dom H = dom ( Y X. Y))
47	46, 17	syl6eq 1944	. . . 4 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> dom dom H = Y)
48	47	raleqdv 2269	. . 3 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> (A.x e. dom dom H((xHU) = x /\ (UHx) = x) <-> A.x e. Y ((xHU) = x /\ (UHx) = x)))
49	42, 48	mpbird 213	. 2 |- ((G e. (Magma i^i ExId ) /\ Y C_ X /\ U e. Y) -> A.x e. dom dom H((xHU) = x /\ (UHx) = x))
50	20, 49	jca 310	1 |- ((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)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105   i^i cin 2592   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987   |` cres 3988  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Idcgi 9312   ExId cexid 10361  Magmacmagm 10365

This theorem is referenced by:  exidres 16031  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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