Theorem grpinvf 8041

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 3355	. . . . . . . . . 10 |- (G e. Grp -> ran G e. V)
2		grpasscan1.1	. . . . . . . . . 10 |- X = ran G
3	1, 2	syl5eqel 1550	. . . . . . . . 9 |- (G e. Grp -> X e. V)
4		rabexg 2720	. . . . . . . . 9 |- (X e. V -> {z e. X | (zGx) = (Id` G)} e. V)
5	3, 4	syl 10	. . . . . . . 8 |- (G e. Grp -> {z e. X | (zGx) = (Id` G)} e. V)
6		uniexg 2867	. . . . . . . 8 |- ({z e. X | (zGx) = (Id` G)} e. V -> U.{z e. X | (zGx) = (Id` G)} e. V)
7	5, 6	syl 10	. . . . . . 7 |- (G e. Grp -> U.{z e. X | (zGx) = (Id` G)} e. V)
8	7	adantr 389	. . . . . 6 |- ((G e. Grp /\ x e. X) -> U.{z e. X | (zGx) = (Id` G)} e. V)
9	8	r19.21aiva 1712	. . . . 5 |- (G e. Grp -> A.x e. X U.{z e. X | (zGx) = (Id` G)} e. V)
10		eqid 1474	. . . . . 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 3612	. . . . 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 198	. . . 4 |- (G e. Grp -> {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} Fn X)
13		eqid 1474	. . . . . 6 |- (Id` G) = (Id` G)
14		grpasscan1.2	. . . . . 6 |- N = (inv` G)
15	2, 13, 14	grpinvfval 8028	. . . . 5 |- (G e. Grp -> N = {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})})
16		fneq1 3578	. . . . 5 |- (N = {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} -> (N Fn X <-> {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} Fn X))
17	15, 16	syl 10	. . . 4 |- (G e. Grp -> (N Fn X <-> {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} Fn X))
18	12, 17	mpbird 196	. . 3 |- (G e. Grp -> N Fn X)
19		fnrnfv 3754	. . . . 5 |- (N Fn X -> ran N = {y | E.x e. X y = (N` x)})
20	18, 19	syl 10	. . . 4 |- (G e. Grp -> ran N = {y | E.x e. X y = (N` x)})
21		fveq2 3719	. . . . . . . . . 10 |- (x = (N` y) -> (N` x) = (N` (N` y)))
22	21	eqeq2d 1484	. . . . . . . . 9 |- (x = (N` y) -> (y = (N` x) <-> y = (N` (N` y))))
23	22	rcla4ev 1874	. . . . . . . 8 |- (((N` y) e. X /\ y = (N` (N` y))) -> E.x e. X y = (N` x))
24	2, 14	grpinvcl 8030	. . . . . . . 8 |- ((G e. Grp /\ y e. X) -> (N` y) e. X)
25	2, 14	grp2inv 8040	. . . . . . . . 9 |- ((G e. Grp /\ y e. X) -> (N` (N` y)) = y)
26	25	eqcomd 1478	. . . . . . . 8 |- ((G e. Grp /\ y e. X) -> y = (N` (N` y)))
27	23, 24, 26	sylanc 471	. . . . . . 7 |- ((G e. Grp /\ y e. X) -> E.x e. X y = (N` x))
28	27	ex 373	. . . . . 6 |- (G e. Grp -> (y e. X -> E.x e. X y = (N` x)))
29		pm3.27 323	. . . . . . . . 9 |- (((G e. Grp /\ x e. X) /\ y = (N` x)) -> y = (N` x))
30	2, 14	grpinvcl 8030	. . . . . . . . . 10 |- ((G e. Grp /\ x e. X) -> (N` x) e. X)
31	30	adantr 389	. . . . . . . . 9 |- (((G e. Grp /\ x e. X) /\ y = (N` x)) -> (N` x) e. X)
32	29, 31	eqeltrd 1546	. . . . . . . 8 |- (((G e. Grp /\ x e. X) /\ y = (N` x)) -> y e. X)
33	32	exp31 376	. . . . . . 7 |- (G e. Grp -> (x e. X -> (y = (N` x) -> y e. X)))
34	33	r19.23adv 1744	. . . . . 6 |- (G e. Grp -> (E.x e. X y = (N` x) -> y e. X))
35	28, 34	impbid 515	. . . . 5 |- (G e. Grp -> (y e. X <-> E.x e. X y = (N` x)))
36	35	abbi2dv 1576	. . . 4 |- (G e. Grp -> X = {y | E.x e. X y = (N` x)})
37	20, 36	eqtr4d 1508	. . 3 |- (G e. Grp -> ran N = X)
38	2, 14	grp2inv 8040	. . . . . . . 8 |- ((G e. Grp /\ x e. X) -> (N` (N` x)) = x)
39	38, 25	eqeqan12d 1488	. . . . . . 7 |- (((G e. Grp /\ x e. X) /\ (G e. Grp /\ y e. X)) -> ((N` (N` x)) = (N` (N` y)) <-> x = y))
40	39	anandis 512	. . . . . 6 |- ((G e. Grp /\ (x e. X /\ y e. X)) -> ((N` (N` x)) = (N` (N` y)) <-> x = y))
41		fveq2 3719	. . . . . 6 |- ((N` x) = (N` y) -> (N` (N` x)) = (N` (N` y)))
42	40, 41	syl5bi 208	. . . . 5 |- ((G e. Grp /\ (x e. X /\ y e. X)) -> ((N` x) = (N` y) -> x = y))
43	42	ex 373	. . . 4 |- (G e. Grp -> ((x e. X /\ y e. X) -> ((N` x) = (N` y) -> x = y)))
44	43	r19.21aivv 1718	. . 3 |- (G e. Grp -> A.x e. X A.y e. X ((N` x) = (N` y) -> x = y))
45	18, 37, 44	3jca 818	. 2 |- (G e. Grp -> (N Fn X /\ ran N = X /\ A.x e. X A.y e. X ((N` x) = (N` y) -> x = y)))
46		f1ofv 3872	. 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)))
47	45, 46	sylibr 200	1 |- (G e. Grp -> N:X-1-1-onto->X)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  {cab 1462  A.wral 1643  E.wrex 1644  {crab 1646  Vcvv 1808  U.cuni 2499  {copab 2662  ran crn 3167   Fn wfn 3173  -1-1-onto->wf1o 3177  ` cfv 3178  (class class class)co 3958  Grpcgr 7995  Idcgi 7996  invcgn 7997

This theorem is referenced by:  invfval 8225

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-opr 3960  df-grp 7999  df-gid 8000  df-ginv 8001

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