Theorem trran2 14757

Description: The range of a right translation.

Hypotheses

Ref	Expression
trfun.2	|- F = (x e. X |-> (xGA))
trinv.1	|- X = ran G

Assertion

Ref	Expression
trran2	|- ((G e. Grp /\ A e. X) -> ran F = X)

Distinct variable groups:   x,A   x,G   x,X

Proof of Theorem trran2

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . . . 7 |- (y = (xGA) -> (y e. X <-> (xGA) e. X))
2		trinv.1	. . . . . . . . . . . 12 |- X = ran G
3	2	grpcl 9324	. . . . . . . . . . 11 |- ((G e. Grp /\ x e. X /\ A e. X) -> (xGA) e. X)
4	3	3exp 1066	. . . . . . . . . 10 |- (G e. Grp -> (x e. X -> (A e. X -> (xGA) e. X)))
5	4	com23 36	. . . . . . . . 9 |- (G e. Grp -> (A e. X -> (x e. X -> (xGA) e. X)))
6	5	imp 377	. . . . . . . 8 |- ((G e. Grp /\ A e. X) -> (x e. X -> (xGA) e. X))
7	6	impcom 378	. . . . . . 7 |- ((x e. X /\ (G e. Grp /\ A e. X)) -> (xGA) e. X)
8	1, 7	syl5cbir 228	. . . . . 6 |- ((x e. X /\ (G e. Grp /\ A e. X)) -> (y = (xGA) -> y e. X))
9	8	expcom 403	. . . . 5 |- ((G e. Grp /\ A e. X) -> (x e. X -> (y = (xGA) -> y e. X)))
10	9	r19.23adv 2215	. . . 4 |- ((G e. Grp /\ A e. X) -> (E.x e. X y = (xGA) -> y e. X))
11		simpll 448	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ y e. X) -> G e. Grp)
12		simpr 350	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ y e. X) -> y e. X)
13		eqid 1884	. . . . . . . . 9 |- (inv` G) = (inv` G)
14	2, 13	grpinvcl 9352	. . . . . . . 8 |- ((G e. Grp /\ A e. X) -> ((inv` G)` A) e. X)
15	14	adantr 425	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ y e. X) -> ((inv` G)` A) e. X)
16	2	grpcl 9324	. . . . . . 7 |- ((G e. Grp /\ y e. X /\ ((inv` G)` A) e. X) -> (yG((inv` G)` A)) e. X)
17	11, 12, 15, 16	syl111anc 1100	. . . . . 6 |- (((G e. Grp /\ A e. X) /\ y e. X) -> (yG((inv` G)` A)) e. X)
18		eqid 1884	. . . . . . . . . . 11 |- (Id` G) = (Id` G)
19	2, 18, 13	grplinv 9354	. . . . . . . . . 10 |- ((G e. Grp /\ A e. X) -> (((inv` G)` A)GA) = (Id` G))
20	19	adantr 425	. . . . . . . . 9 |- (((G e. Grp /\ A e. X) /\ y e. X) -> (((inv` G)` A)GA) = (Id` G))
21	20	eqcomd 1889	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ y e. X) -> (Id` G) = (((inv` G)` A)GA))
22	21	opreq2d 4898	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ y e. X) -> (yG(Id` G)) = (yG(((inv` G)` A)GA)))
23	2, 18	grprid 9346	. . . . . . . . 9 |- ((G e. Grp /\ y e. X) -> (yG(Id` G)) = y)
24	23	eqcomd 1889	. . . . . . . 8 |- ((G e. Grp /\ y e. X) -> y = (yG(Id` G)))
25	24	adantlr 429	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ y e. X) -> y = (yG(Id` G)))
26		simplr 449	. . . . . . . 8 |- (((G e. Grp /\ A e. X) /\ y e. X) -> A e. X)
27	2	grpass 9327	. . . . . . . 8 |- ((G e. Grp /\ (y e. X /\ ((inv` G)` A) e. X /\ A e. X)) -> ((yG((inv` G)` A))GA) = (yG(((inv` G)` A)GA)))
28	11, 12, 15, 26, 27	syl13anc 1102	. . . . . . 7 |- (((G e. Grp /\ A e. X) /\ y e. X) -> ((yG((inv` G)` A))GA) = (yG(((inv` G)` A)GA)))
29	22, 25, 28	3eqtr4d 1937	. . . . . 6 |- (((G e. Grp /\ A e. X) /\ y e. X) -> y = ((yG((inv` G)` A))GA))
30		opreq1 4889	. . . . . . . 8 |- (x = (yG((inv` G)` A)) -> (xGA) = ((yG((inv` G)` A))GA))
31	30	eqeq2d 1895	. . . . . . 7 |- (x = (yG((inv` G)` A)) -> (y = (xGA) <-> y = ((yG((inv` G)` A))GA)))
32	31	rcla4ev 2381	. . . . . 6 |- (((yG((inv` G)` A)) e. X /\ y = ((yG((inv` G)` A))GA)) -> E.x e. X y = (xGA))
33	17, 29, 32	syl11anc 524	. . . . 5 |- (((G e. Grp /\ A e. X) /\ y e. X) -> E.x e. X y = (xGA))
34	33	ex 402	. . . 4 |- ((G e. Grp /\ A e. X) -> (y e. X -> E.x e. X y = (xGA)))
35	10, 34	impbid 574	. . 3 |- ((G e. Grp /\ A e. X) -> (E.x e. X y = (xGA) <-> y e. X))
36	35	abbi1dv 2010	. 2 |- ((G e. Grp /\ A e. X) -> {y | E.x e. X y = (xGA)} = X)
37		trfun.2	. . 3 |- F = (x e. X |-> (xGA))
38	37	cmpran 14483	. 2 |- ran F = {y | E.x e. X y = (xGA)}
39	36, 38	syl5eq 1940	1 |- ((G e. Grp /\ A e. X) -> ran F = X)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106  ran crn 3987  ` cfv 3998  (class class class)co 4884   e. cmpt 5004  Grpcgr 9311  Idcgi 9312  invcgn 9313

This theorem is referenced by:  trooo 14758

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-mpt 5006  df-grp 9316  df-gid 9317  df-ginv 9318

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