Theorem grpinvf 8079

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 3359	. . . . . . . . . 10 |- (G e. Grp -> ran G e. V)
2		grpasscan1.1	. . . . . . . . . 10 |- X = ran G
3	1, 2	syl5eqel 1552	. . . . . . . . 9 |- (G e. Grp -> X e. V)
4		rabexg 2724	. . . . . . . . 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 2871	. . . . . . . 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 1714	. . . . 5 |- (G e. Grp -> A.x e. X U.{z e. X | (zGx) = (Id` G)} e. V)
10		eqid 1475	. . . . . 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 3616	. . . . 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 1475	. . . . . 6 |- (Id` G) = (Id` G)
14		grpasscan1.2	. . . . . 6 |- N = (inv` G)
15	2, 13, 14	grpinvfval 8066	. . . . 5 |- (G e. Grp -> N = {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})})
16		fneq1 3582	. . . . 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 3759	. . . . 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 3724	. . . . . . . . . 10 |- (x = (N` y) -> (N` x) = (N` (N` y)))
22	21	eqeq2d 1486	. . . . . . . . 9 |- (x = (N` y) -> (y = (N` x) <-> y = (N` (N` y))))
23	22	rcla4ev 1877	. . . . . . . 8 |- (((N` y) e. X /\ y = (N` (N` y))) -> E.x e. X y = (N` x))
24	2, 14	grpinvcl 8068	. . . . . . . 8 |- ((G e. Grp /\ y e. X) -> (N` y) e. X)
25	2, 14	grp2inv 8078	. . . . . . . . 9 |- ((G e. Grp /\ y e. X) -> (N` (N` y)) = y)
26	25	eqcomd 1480	. . . . . . . 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 8068	. . . . . . . . . 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 1548	. . . . . . . 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 1746	. . . . . 6 |- (G e. Grp -> (E.x e. X y = (N` x) -> y e. X))
35	28, 34	impbid 516	. . . . 5 |- (G e. Grp -> (y e. X <-> E.x e. X y = (N` x)))
36	35	abbi2dv 1578	. . . 4 |- (G e. Grp -> X = {y | E.x e. X y = (N` x)})
37	20, 36	eqtr4d 1510	. . 3 |- (G e. Grp -> ran N = X)
38	2, 14	grp2inv 8078	. . . . . . . 8 |- ((G e. Grp /\ x e. X) -> (N` (N` x)) = x)
39	38, 25	eqeqan12d 1490	. . . . . . 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 3724	. . . . . 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 1720	. . 3 |- (G e. Grp -> A.x e. X A.y e. X ((N` x) = (N` y) -> x = y))
45	18, 37, 44	3jca 819	. 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 3877	. 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 775   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646  {crab 1648  Vcvv 1811  U.cuni 2503  {copab 2666  ran crn 3171   Fn wfn 3177  -1-1-onto->wf1o 3181  ` cfv 3182  (class class class)co 3963  Grpcgr 8033  Idcgi 8034  invcgn 8035

This theorem is referenced by:  invfval 8261

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-grp 8037  df-gid 8038  df-ginv 8039

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