Theorem grpdivfo 14737

Description: A "division" maps onto the group's underlying set.

Hypotheses

Ref	Expression
grpdivfo.1	|- X = ran G
grpdivfo.2	|- D = ( /g ` G)

Assertion

Ref	Expression
grpdivfo	|- (G e. Grp -> D:(X X. X)-onto->X)

Proof of Theorem grpdivfo

Step	Hyp	Ref	Expression
1		grpdivfo.1	. . . 4 |- X = ran G
2		grpdivfo.2	. . . 4 |- D = ( /g ` G)
3	1, 2	grpdivf 9370	. . 3 |- (G e. Grp -> D:(X X. X)-->X)
4		ffn 4562	. . . . . 6 |- (D:(X X. X)-->X -> D Fn (X X. X))
5	3, 4	syl 12	. . . . 5 |- (G e. Grp -> D Fn (X X. X))
6		fnrnoprv 4966	. . . . 5 |- (D Fn (X X. X) -> ran D = {x | E.y e. X E.z e. X x = (yDz)})
7	5, 6	syl 12	. . . 4 |- (G e. Grp -> ran D = {x | E.y e. X E.z e. X x = (yDz)})
8		foprrn 4965	. . . . . . . . . . . . . . . . 17 |- ((D:(X X. X)-->X /\ y e. X /\ z e. X) -> (yDz) e. X)
9		eleq1 1957	. . . . . . . . . . . . . . . . . . . 20 |- ((yDz) = x -> ((yDz) e. X <-> x e. X))
10	9	biimpd 170	. . . . . . . . . . . . . . . . . . 19 |- ((yDz) = x -> ((yDz) e. X -> x e. X))
11	10	eqcoms 1887	. . . . . . . . . . . . . . . . . 18 |- (x = (yDz) -> ((yDz) e. X -> x e. X))
12	11	com12 14	. . . . . . . . . . . . . . . . 17 |- ((yDz) e. X -> (x = (yDz) -> x e. X))
13	8, 12	syl 12	. . . . . . . . . . . . . . . 16 |- ((D:(X X. X)-->X /\ y e. X /\ z e. X) -> (x = (yDz) -> x e. X))
14	13	3exp 1066	. . . . . . . . . . . . . . 15 |- (D:(X X. X)-->X -> (y e. X -> (z e. X -> (x = (yDz) -> x e. X))))
15	3, 14	syl 12	. . . . . . . . . . . . . 14 |- (G e. Grp -> (y e. X -> (z e. X -> (x = (yDz) -> x e. X))))
16	15	com3l 38	. . . . . . . . . . . . 13 |- (y e. X -> (z e. X -> (G e. Grp -> (x = (yDz) -> x e. X))))
17	16	com34 40	. . . . . . . . . . . 12 |- (y e. X -> (z e. X -> (x = (yDz) -> (G e. Grp -> x e. X))))
18	17	com3l 38	. . . . . . . . . . 11 |- (z e. X -> (x = (yDz) -> (y e. X -> (G e. Grp -> x e. X))))
19	18	r19.23aiv 2211	. . . . . . . . . 10 |- (E.z e. X x = (yDz) -> (y e. X -> (G e. Grp -> x e. X)))
20	19	com12 14	. . . . . . . . 9 |- (y e. X -> (E.z e. X x = (yDz) -> (G e. Grp -> x e. X)))
21	20	r19.23aiv 2211	. . . . . . . 8 |- (E.y e. X E.z e. X x = (yDz) -> (G e. Grp -> x e. X))
22	21	com12 14	. . . . . . 7 |- (G e. Grp -> (E.y e. X E.z e. X x = (yDz) -> x e. X))
23		eqid 1884	. . . . . . . . . 10 |- (Id` G) = (Id` G)
24	1, 23	grpidcl 9343	. . . . . . . . 9 |- (G e. Grp -> (Id` G) e. X)
25	1, 2, 23	grpdivone 14736	. . . . . . . . . . . . 13 |- ((G e. Grp /\ x e. X) -> (xD(Id` G)) = x)
26		rcla4eopr 4915	. . . . . . . . . . . . . . . . 17 |- ((x e. X /\ (Id` G) e. X /\ x = (xD(Id` G))) -> E.y e. X E.z e. X x = (yDz))
27	26	3exp 1066	. . . . . . . . . . . . . . . 16 |- (x e. X -> ((Id` G) e. X -> (x = (xD(Id` G)) -> E.y e. X E.z e. X x = (yDz))))
28	27	adantl 424	. . . . . . . . . . . . . . 15 |- ((G e. Grp /\ x e. X) -> ((Id` G) e. X -> (x = (xD(Id` G)) -> E.y e. X E.z e. X x = (yDz))))
29	28	com3r 39	. . . . . . . . . . . . . 14 |- (x = (xD(Id` G)) -> ((G e. Grp /\ x e. X) -> ((Id` G) e. X -> E.y e. X E.z e. X x = (yDz))))
30	29	eqcoms 1887	. . . . . . . . . . . . 13 |- ((xD(Id` G)) = x -> ((G e. Grp /\ x e. X) -> ((Id` G) e. X -> E.y e. X E.z e. X x = (yDz))))
31	25, 30	syl 12	. . . . . . . . . . . 12 |- ((G e. Grp /\ x e. X) -> ((G e. Grp /\ x e. X) -> ((Id` G) e. X -> E.y e. X E.z e. X x = (yDz))))
32	31	pm2.43i 78	. . . . . . . . . . 11 |- ((G e. Grp /\ x e. X) -> ((Id` G) e. X -> E.y e. X E.z e. X x = (yDz)))
33	32	ex 402	. . . . . . . . . 10 |- (G e. Grp -> (x e. X -> ((Id` G) e. X -> E.y e. X E.z e. X x = (yDz))))
34	33	com3r 39	. . . . . . . . 9 |- ((Id` G) e. X -> (G e. Grp -> (x e. X -> E.y e. X E.z e. X x = (yDz))))
35	24, 34	syl 12	. . . . . . . 8 |- (G e. Grp -> (G e. Grp -> (x e. X -> E.y e. X E.z e. X x = (yDz))))
36	35	pm2.43i 78	. . . . . . 7 |- (G e. Grp -> (x e. X -> E.y e. X E.z e. X x = (yDz)))
37	22, 36	impbid 574	. . . . . 6 |- (G e. Grp -> (E.y e. X E.z e. X x = (yDz) <-> x e. X))
38	37	19.21aiv 1664	. . . . 5 |- (G e. Grp -> A.x(E.y e. X E.z e. X x = (yDz) <-> x e. X))
39		abeq1 2000	. . . . . 6 |- ({x | E.y e. X E.z e. X x = (yDz)} = X <-> A.x(E.y e. X E.z e. X x = (yDz) <-> x e. X))
40	39	a1i 8	. . . . 5 |- (G e. Grp -> ({x | E.y e. X E.z e. X x = (yDz)} = X <-> A.x(E.y e. X E.z e. X x = (yDz) <-> x e. X)))
41	38, 40	mpbird 213	. . . 4 |- (G e. Grp -> {x | E.y e. X E.z e. X x = (yDz)} = X)
42	7, 41	eqtrd 1925	. . 3 |- (G e. Grp -> ran D = X)
43	3, 42	jca 310	. 2 |- (G e. Grp -> (D:(X X. X)-->X /\ ran D = X))
44		dffo2 4621	. . 3 |- (D:(X X. X)-onto->X <-> (D:(X X. X)-->X /\ ran D = X))
45	44	a1i 8	. 2 |- (G e. Grp -> (D:(X X. X)-onto->X <-> (D:(X X. X)-->X /\ ran D = X)))
46	43, 45	mpbird 213	1 |- (G e. Grp -> D:(X X. X)-onto->X)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106   X. cxp 3984  ran crn 3987   Fn wfn 3993  -->wf 3994  -onto->wfo 3996  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312   /g cgs 9314

This theorem is referenced by:  trhom 14983

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-rep 3428  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-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319

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