Theorem grpinvf 8198

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 3419	. . . . . . . . . 10 |- (G e. Grp -> ran G e. V)
2		grpasscan1.1	. . . . . . . . . 10 |- X = ran G
3	1, 2	syl5eqel 1589	. . . . . . . . 9 |- (G e. Grp -> X e. V)
4		rabexg 2775	. . . . . . . . 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 2925	. . . . . . . 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 1752	. . . . 5 |- (G e. Grp -> A.x e. X U.{z e. X | (zGx) = (Id` G)} e. V)
10		eqid 1512	. . . . . 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 3691	. . . . 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 196	. . . 4 |- (G e. Grp -> {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})} Fn X)
13		eqid 1512	. . . . . 6 |- (Id` G) = (Id` G)
14		grpasscan1.2	. . . . . 6 |- N = (inv` G)
15	2, 13, 14	grpinvfval 8185	. . . . 5 |- (G e. Grp -> N = {<.x, y>. | (x e. X /\ y = U.{z e. X | (zGx) = (Id` G)})})
16		fneq1 3657	. . . . 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 194	. . 3 |- (G e. Grp -> N Fn X)
19		fnrnfv 3835	. . . . 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 3800	. . . . . . . . . 10 |- (x = (N` y) -> (N` x) = (N` (N` y)))
22	21	eqeq2d 1523	. . . . . . . . 9 |- (x = (N` y) -> (y = (N` x) <-> y = (N` (N` y))))
23	22	rcla4ev 1915	. . . . . . . 8 |- (((N` y) e. X /\ y = (N` (N` y))) -> E.x e. X y = (N` x))
24	2, 14	grpinvcl 8187	. . . . . . . 8 |- ((G e. Grp /\ y e. X) -> (N` y) e. X)
25	2, 14	grp2inv 8197	. . . . . . . . 9 |- ((G e. Grp /\ y e. X) -> (N` (N` y)) = y)
26	25	eqcomd 1517	. . . . . . . 8 |- ((G e. Grp /\ y e. X) -> y = (N` (N` y)))
27	23, 24, 26	sylanc 473	. . . . . . 7 |- ((G e. Grp /\ y e. X) -> E.x e. X y = (N` x))
28	27	ex 371	. . . . . 6 |- (G e. Grp -> (y e. X -> E.x e. X y = (N` x)))
29		pm3.27 321	. . . . . . . . 9 |- (((G e. Grp /\ x e. X) /\ y = (N` x)) -> y = (N` x))
30	2, 14	grpinvcl 8187	. . . . . . . . . 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 1585	. . . . . . . 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 1784	. . . . . 6 |- (G e. Grp -> (E.x e. X y = (N` x) -> y e. X))
35	28, 34	impbid 518	. . . . 5 |- (G e. Grp -> (y e. X <-> E.x e. X y = (N` x)))
36	35	abbi2dv 1615	. . . 4 |- (G e. Grp -> X = {y | E.x e. X y = (N` x)})
37	20, 36	eqtr4d 1547	. . 3 |- (G e. Grp -> ran N = X)
38	2, 14	grp2inv 8197	. . . . . . . 8 |- ((G e. Grp /\ x e. X) -> (N` (N` x)) = x)
39	38, 25	eqeqan12d 1527	. . . . . . 7 |- (((G e. Grp /\ x e. X) /\ (G e. Grp /\ y e. X)) -> ((N` (N` x)) = (N` (N` y)) <-> x = y))
40	39	anandis 514	. . . . . 6 |- ((G e. Grp /\ (x e. X /\ y e. X)) -> ((N` (N` x)) = (N` (N` y)) <-> x = y))
41		fveq2 3800	. . . . . 6 |- ((N` x) = (N` y) -> (N` (N` x)) = (N` (N` y)))
42	40, 41	syl5bi 206	. . . . 5 |- ((G e. Grp /\ (x e. X /\ y e. X)) -> ((N` x) = (N` y) -> x = y))
43	42	ex 371	. . . 4 |- (G e. Grp -> ((x e. X /\ y e. X) -> ((N` x) = (N` y) -> x = y)))
44	43	r19.21aivv 1758	. . 3 |- (G e. Grp -> A.x e. X A.y e. X ((N` x) = (N` y) -> x = y))
45	18, 37, 44	3jca 822	. 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		dff1o6 3953	. 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 198	1 |- (G e. Grp -> N:X-1-1-onto->X)

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   /\ w3a 778   = wceq 988   e. wcel 990  {cab 1499  A.wral 1683  E.wrex 1684  {crab 1686  Vcvv 1849  U.cuni 2551  {copab 2717  ran crn 3226   Fn wfn 3232  -1-1-onto->wf1o 3236  ` cfv 3237  (class class class)co 4039  Grpcgr 8153  Idcgi 8154  invcgn 8155

This theorem is referenced by:  invfval 8380

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-reu 1689  df-rab 1690  df-v 1850  df-sbc 1979  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253  df-opr 4041  df-grp 8157  df-gid 8158  df-ginv 8159

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