Theorem releqgg 13290

Description: The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypothesis

Ref	Expression
releqg.r	|- R = ( G ~QG S )

Assertion

Ref	Expression
releqgg	|- ( ( G e. V /\ S e. W ) -> Rel R )

Proof of Theorem releqgg

Dummy variables i r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 4788	. 2 |- Rel { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) }
2		releqg.r	. . . 4 |- R = ( G ~QG S )
3		elex 2771	. . . . . 6 |- ( G e. V -> G e. _V )
4	3	adantr 276	. . . . 5 |- ( ( G e. V /\ S e. W ) -> G e. _V )
5		elex 2771	. . . . . 6 |- ( S e. W -> S e. _V )
6	5	adantl 277	. . . . 5 |- ( ( G e. V /\ S e. W ) -> S e. _V )
7		vex 2763	. . . . . . . . 9 |- x e. _V
8		vex 2763	. . . . . . . . 9 |- y e. _V
9	7, 8	prss 3774	. . . . . . . 8 |- ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) <-> { x , y } C_ ( Base ` G ) )
10	9	anbi1i 458	. . . . . . 7 |- ( ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) <-> ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) )
11	10	opabbii 4096	. . . . . 6 |- { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) }
12		basfn 12676	. . . . . . . . 9 |- Base Fn _V
13		funfvex 5571	. . . . . . . . . 10 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
14	13	funfni 5354	. . . . . . . . 9 |- ( ( Base Fn _V /\ G e. _V ) -> ( Base ` G ) e. _V )
15	12, 4, 14	sylancr 414	. . . . . . . 8 |- ( ( G e. V /\ S e. W ) -> ( Base ` G ) e. _V )
16		xpexg 4773	. . . . . . . 8 |- ( ( ( Base ` G ) e. _V /\ ( Base ` G ) e. _V ) -> ( ( Base ` G ) X. ( Base ` G ) ) e. _V )
17	15, 15, 16	syl2anc 411	. . . . . . 7 |- ( ( G e. V /\ S e. W ) -> ( ( Base ` G ) X. ( Base ` G ) ) e. _V )
18		opabssxp 4733	. . . . . . . 8 |- { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } C_ ( ( Base ` G ) X. ( Base ` G ) )
19	18	a1i 9	. . . . . . 7 |- ( ( G e. V /\ S e. W ) -> { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } C_ ( ( Base ` G ) X. ( Base ` G ) ) )
20	17, 19	ssexd 4169	. . . . . 6 |- ( ( G e. V /\ S e. W ) -> { <. x , y >. | ( ( x e. ( Base ` G ) /\ y e. ( Base ` G ) ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } e. _V )
21	11, 20	eqeltrrid 2281	. . . . 5 |- ( ( G e. V /\ S e. W ) -> { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } e. _V )
22		fveq2 5554	. . . . . . . . 9 |- ( r = G -> ( Base ` r ) = ( Base ` G ) )
23	22	sseq2d 3209	. . . . . . . 8 |- ( r = G -> ( { x , y } C_ ( Base ` r ) <-> { x , y } C_ ( Base ` G ) ) )
24		fveq2 5554	. . . . . . . . . 10 |- ( r = G -> ( +g ` r ) = ( +g ` G ) )
25		fveq2 5554	. . . . . . . . . . 11 |- ( r = G -> ( invg ` r ) = ( invg ` G ) )
26	25	fveq1d 5556	. . . . . . . . . 10 |- ( r = G -> ( ( invg ` r ) ` x ) = ( ( invg ` G ) ` x ) )
27		eqidd 2194	. . . . . . . . . 10 |- ( r = G -> y = y )
28	24, 26, 27	oveq123d 5939	. . . . . . . . 9 |- ( r = G -> ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) = ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) )
29	28	eleq1d 2262	. . . . . . . 8 |- ( r = G -> ( ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i <-> ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) )
30	23, 29	anbi12d 473	. . . . . . 7 |- ( r = G -> ( ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) <-> ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) ) )
31	30	opabbidv 4095	. . . . . 6 |- ( r = G -> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) } = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) } )
32		eleq2 2257	. . . . . . . 8 |- ( i = S -> ( ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i <-> ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) )
33	32	anbi2d 464	. . . . . . 7 |- ( i = S -> ( ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) <-> ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) ) )
34	33	opabbidv 4095	. . . . . 6 |- ( i = S -> { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. i ) } = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
35		df-eqg 13242	. . . . . 6 |- ~QG = ( r e. _V , i e. _V |-> { <. x , y >. | ( { x , y } C_ ( Base ` r ) /\ ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) e. i ) } )
36	31, 34, 35	ovmpog 6053	. . . . 5 |- ( ( G e. _V /\ S e. _V /\ { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } e. _V ) -> ( G ~QG S ) = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
37	4, 6, 21, 36	syl3anc 1249	. . . 4 |- ( ( G e. V /\ S e. W ) -> ( G ~QG S ) = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
38	2, 37	eqtrid 2238	. . 3 |- ( ( G e. V /\ S e. W ) -> R = { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } )
39	38	releqd 4743	. 2 |- ( ( G e. V /\ S e. W ) -> ( Rel R <-> Rel { <. x , y >. | ( { x , y } C_ ( Base ` G ) /\ ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) e. S ) } ) )
40	1, 39	mpbiri 168	1 |- ( ( G e. V /\ S e. W ) -> Rel R )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   {cpr 3619   {copab 4089   X. cxp 4657   Rel wrel 4664   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   invgcminusg 13073   ~_QG cqg 13239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-inn 8983  df-ndx 12621  df-slot 12622  df-base 12624  df-eqg 13242

This theorem is referenced by:  eqger  13294  eqgid  13296