Theorem prsubrtr 14763

Description: The product of a subset B of X by an element of X is the image of B by a right translation.

Hypotheses

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

Assertion

Ref	Expression
prsubrtr	|- ((G e. Grp /\ A e. X /\ B e. ~PX) -> (BH{A}) = (F"B))

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

Proof of Theorem prsubrtr

Step	Hyp	Ref	Expression
1		snnzg 3118	. . . . . . . . 9 |- (A e. X -> {A} =/= (/))
2	1	3ad2ant2 898	. . . . . . . 8 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> {A} =/= (/))
3	2	adantr 425	. . . . . . 7 |- (((G e. Grp /\ A e. X /\ B e. ~PX) /\ x e. B) -> {A} =/= (/))
4		r19.9rzv 2963	. . . . . . . 8 |- ({A} =/= (/) -> (a = (xGA) <-> E.v e. {A}a = (xGA)))
5	4	bicomd 580	. . . . . . 7 |- ({A} =/= (/) -> (E.v e. {A}a = (xGA) <-> a = (xGA)))
6	3, 5	syl 12	. . . . . 6 |- (((G e. Grp /\ A e. X /\ B e. ~PX) /\ x e. B) -> (E.v e. {A}a = (xGA) <-> a = (xGA)))
7		elsn 3058	. . . . . . . 8 |- (v e. {A} <-> v = A)
8		opreq2 4890	. . . . . . . . 9 |- (v = A -> (xGv) = (xGA))
9	8	eqeq2d 1895	. . . . . . . 8 |- (v = A -> (a = (xGv) <-> a = (xGA)))
10	7, 9	sylbi 216	. . . . . . 7 |- (v e. {A} -> (a = (xGv) <-> a = (xGA)))
11	10	rexbiia 2134	. . . . . 6 |- (E.v e. {A}a = (xGv) <-> E.v e. {A}a = (xGA))
12	6, 11	syl5bb 591	. . . . 5 |- (((G e. Grp /\ A e. X /\ B e. ~PX) /\ x e. B) -> (E.v e. {A}a = (xGv) <-> a = (xGA)))
13		trfun.2	. . . . . . . . 9 |- F = (x e. X |-> (xGA))
14	13	fvopab2b 14476	. . . . . . . 8 |- ((x e. X /\ (xGA) e. _V) -> (F` x) = (xGA))
15		elelpwi 3040	. . . . . . . . . . 11 |- ((x e. B /\ B e. ~PX) -> x e. X)
16	15	expcom 403	. . . . . . . . . 10 |- (B e. ~PX -> (x e. B -> x e. X))
17	16	3ad2ant3 899	. . . . . . . . 9 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> (x e. B -> x e. X))
18	17	imp 377	. . . . . . . 8 |- (((G e. Grp /\ A e. X /\ B e. ~PX) /\ x e. B) -> x e. X)
19		oprex 4907	. . . . . . . 8 |- (xGA) e. _V
20	14, 18, 19	sylancl 525	. . . . . . 7 |- (((G e. Grp /\ A e. X /\ B e. ~PX) /\ x e. B) -> (F` x) = (xGA))
21	20	eqcomd 1889	. . . . . 6 |- (((G e. Grp /\ A e. X /\ B e. ~PX) /\ x e. B) -> (xGA) = (F` x))
22	21	eqeq2d 1895	. . . . 5 |- (((G e. Grp /\ A e. X /\ B e. ~PX) /\ x e. B) -> (a = (xGA) <-> a = (F` x)))
23	12, 22	bitrd 587	. . . 4 |- (((G e. Grp /\ A e. X /\ B e. ~PX) /\ x e. B) -> (E.v e. {A}a = (xGv) <-> a = (F` x)))
24	23	rexbidva 2120	. . 3 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> (E.x e. B E.v e. {A}a = (xGv) <-> E.x e. B a = (F` x)))
25	24	abbidv 2008	. 2 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> {a | E.x e. B E.v e. {A}a = (xGv)} = {a | E.x e. B a = (F` x)})
26		simp1 876	. . 3 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> G e. Grp)
27		grprndm 9334	. . . . . . . 8 |- (G e. Grp -> ran G = dom dom G)
28		trinv.1	. . . . . . . 8 |- X = ran G
29	27, 28	syl5eq 1940	. . . . . . 7 |- (G e. Grp -> X = dom dom G)
30		pweq 3036	. . . . . . 7 |- (X = dom dom G -> ~PX = ~Pdom dom G)
31	29, 30	syl 12	. . . . . 6 |- (G e. Grp -> ~PX = ~Pdom dom G)
32	31	eleq2d 1964	. . . . 5 |- (G e. Grp -> (B e. ~PX <-> B e. ~Pdom dom G))
33	32	biimpa 460	. . . 4 |- ((G e. Grp /\ B e. ~PX) -> B e. ~Pdom dom G)
34	33	3adant2 895	. . 3 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> B e. ~Pdom dom G)
35	29	eleq2d 1964	. . . . . 6 |- (G e. Grp -> (A e. X <-> A e. dom dom G))
36	35	biimpa 460	. . . . 5 |- ((G e. Grp /\ A e. X) -> A e. dom dom G)
37		snelpwg 14415	. . . . . 6 |- (A e. X -> (A e. dom dom G <-> {A} e. ~Pdom dom G))
38	37	adantl 424	. . . . 5 |- ((G e. Grp /\ A e. X) -> (A e. dom dom G <-> {A} e. ~Pdom dom G))
39	36, 38	mpbid 212	. . . 4 |- ((G e. Grp /\ A e. X) -> {A} e. ~Pdom dom G)
40	39	3adant3 896	. . 3 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> {A} e. ~Pdom dom G)
41		eqid 1884	. . . 4 |- dom dom G = dom dom G
42		prsubrtr.1	. . . 4 |- H = (cset` G)
43	41, 42	iscst2 14520	. . 3 |- ((G e. Grp /\ B e. ~Pdom dom G /\ {A} e. ~Pdom dom G) -> (BH{A}) = {a | E.x e. B E.v e. {A}a = (xGv)})
44	26, 34, 40, 43	syl111anc 1100	. 2 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> (BH{A}) = {a | E.x e. B E.v e. {A}a = (xGv)})
45	13, 28	imtr 14762	. 2 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> (F"B) = {a | E.x e. B a = (F` x)})
46	25, 44, 45	3eqtr4d 1937	1 |- ((G e. Grp /\ A e. X /\ B e. ~PX) -> (BH{A}) = (F"B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  E.wrex 2106  _Vcvv 2292  (/)c0 2875  ~Pcpw 3032  {csn 3044  dom cdm 3986  ran crn 3987  "cima 3989  ` cfv 3998  (class class class)co 4884   e. cmpt 5004  Grpcgr 9311  csetccst 14517

This theorem is referenced by:  prsubrtr2 14764

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-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-iun 3257  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-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-grp 9316  df-cst 14518

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