Theorem grpinvf 9364

Description: Mapping of the inverse function of a group.

Hypotheses

Ref	Expression
grpasscan1.1	|- X = ran G
grpasscan1.2	|- N = (inv` G)

Assertion

Ref	Expression
grpinvf	|- (G e. Grp -> N:X-1-1-onto->X)

Proof of Theorem grpinvf

Step	Hyp	Ref	Expression
1		rnexg 4207	. . . . . . . . . 10 |- (G e. Grp -> ran G e. _V)
2		grpasscan1.1	. . . . . . . . . 10 |- X = ran G
3	1, 2	syl5eqel 1975	. . . . . . . . 9 |- (G e. Grp -> X e. _V)
4		rabexg 3460	. . . . . . . . 9 |- (X e. _V -> {z e. X | (zGx) = (Id` G)} e. _V)
5	3, 4	syl 12	. . . . . . . 8 |- (G e. Grp -> {z e. X | (zGx) = (Id` G)} e. _V)
6		uniexg 3795	. . . . . . . 8 |- ({z e. X | (zGx) = (Id` G)} e. _V -> U.{z e. X | (zGx) = (Id` G)} e. _V)
7	5, 6	syl 12	. . . . . . 7 |- (G e. Grp -> U.{z e. X | (zGx) = (Id` G)} e. _V)
8	7	adantr 425	. . . . . 6 |- ((G e. Grp /\ x e. X) -> U.{z e. X | (zGx) = (Id` G)} e. _V)
9	8	r19.21aiva 2176	. . . . 5 |- (G e. Grp -> A.x e. X U.{z e. X | (zGx) = (Id` G)} e. _V)
10		eqid 1884	. . . . . 6 |- {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} = {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})}
11	10	fnopab2g 4547	. . . . 5 |- (A.x e. X U.{z e. X | (zGx) = (Id` G)} e. _V <-> {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} Fn X)
12	9, 11	sylib 215	. . . 4 |- (G e. Grp -> {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} Fn X)
13		eqid 1884	. . . . . 6 |- (Id` G) = (Id` G)
14		grpasscan1.2	. . . . . 6 |- N = (inv` G)
15	2, 13, 14	grpinvfval 9350	. . . . 5 |- (G e. Grp -> N = {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})})
16	15	fneq1d 4505	. . . 4 |- (G e. Grp -> (N Fn X <-> {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} Fn X))
17	12, 16	mpbird 213	. . 3 |- (G e. Grp -> N Fn X)
18		fnrnfv 4718	. . . . 5 |- (N Fn X -> ran N = {y | E.x e. X y = (N` x)})
19	17, 18	syl 12	. . . 4 |- (G e. Grp -> ran N = {y | E.x e. X y = (N` x)})
20	2, 14	grpinvcl 9352	. . . . . . . 8 |- ((G e. Grp /\ y e. X) -> (N` y) e. X)
21	2, 14	grp2inv 9363	. . . . . . . . 9 |- ((G e. Grp /\ y e. X) -> (N` (N` y)) = y)
22	21	eqcomd 1889	. . . . . . . 8 |- ((G e. Grp /\ y e. X) -> y = (N` (N` y)))
23		fveq2 4681	. . . . . . . . . 10 |- (x = (N` y) -> (N` x) = (N` (N` y)))
24	23	eqeq2d 1895	. . . . . . . . 9 |- (x = (N` y) -> (y = (N` x) <-> y = (N` (N` y))))
25	24	rcla4ev 2381	. . . . . . . 8 |- (((N` y) e. X /\ y = (N` (N` y))) -> E.x e. X y = (N` x))
26	20, 22, 25	syl11anc 524	. . . . . . 7 |- ((G e. Grp /\ y e. X) -> E.x e. X y = (N` x))
27	26	ex 402	. . . . . 6 |- (G e. Grp -> (y e. X -> E.x e. X y = (N` x)))
28		simpr 350	. . . . . . . . 9 |- (((G e. Grp /\ x e. X) /\ y = (N` x)) -> y = (N` x))
29	2, 14	grpinvcl 9352	. . . . . . . . . 10 |- ((G e. Grp /\ x e. X) -> (N` x) e. X)
30	29	adantr 425	. . . . . . . . 9 |- (((G e. Grp /\ x e. X) /\ y = (N` x)) -> (N` x) e. X)
31	28, 30	eqeltrd 1971	. . . . . . . 8 |- (((G e. Grp /\ x e. X) /\ y = (N` x)) -> y e. X)
32	31	exp31 407	. . . . . . 7 |- (G e. Grp -> (x e. X -> (y = (N` x) -> y e. X)))
33	32	r19.23adv 2215	. . . . . 6 |- (G e. Grp -> (E.x e. X y = (N` x) -> y e. X))
34	27, 33	impbid 574	. . . . 5 |- (G e. Grp -> (y e. X <-> E.x e. X y = (N` x)))
35	34	abbi2dv 2009	. . . 4 |- (G e. Grp -> X = {y | E.x e. X y = (N` x)})
36	19, 35	eqtr4d 1928	. . 3 |- (G e. Grp -> ran N = X)
37	2, 14	grp2inv 9363	. . . . . . . 8 |- ((G e. Grp /\ x e. X) -> (N` (N` x)) = x)
38	37, 21	eqeqan12d 1901	. . . . . . 7 |- (((G e. Grp /\ x e. X) /\ (G e. Grp /\ y e. X)) -> ((N` (N` x)) = (N` (N` y)) <-> x = y))
39	38	anandis 570	. . . . . 6 |- ((G e. Grp /\ (x e. X /\ y e. X)) -> ((N` (N` x)) = (N` (N` y)) <-> x = y))
40		fveq2 4681	. . . . . 6 |- ((N` x) = (N` y) -> (N` (N` x)) = (N` (N` y)))
41	39, 40	syl5bi 225	. . . . 5 |- ((G e. Grp /\ (x e. X /\ y e. X)) -> ((N` x) = (N` y) -> x = y))
42	41	ex 402	. . . 4 |- (G e. Grp -> ((x e. X /\ y e. X) -> ((N` x) = (N` y) -> x = y)))
43	42	r19.21aivv 2183	. . 3 |- (G e. Grp -> A.x e. X A.y e. X ((N` x) = (N` y) -> x = y))
44	17, 36, 43	3jca 1050	. 2 |- (G e. Grp -> (N Fn X /\ ran N = X /\ A.x e. X A.y e. X ((N` x) = (N` y) -> x = y)))
45		dff1o6 4853	. 2 |- (N:X-1-1-onto->X <-> (N Fn X /\ ran N = X /\ A.x e. X A.y e. X ((N` x) = (N` y) -> x = y)))
46	44, 45	sylibr 217	1 |- (G e. Grp -> N:X-1-1-onto->X)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292  U.cuni 3177  {copab 3395  ran crn 3987   Fn wfn 3993  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  Grpcgr 9311  Idcgi 9312  invcgn 9313

This theorem is referenced by:  invfval 9593  grpdlcan 14739  grpdivzer 14740  topgrpbs 14974

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-grp 9316  df-gid 9317  df-ginv 9318

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