Theorem releqgg 13085

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 4755	. 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 2750	. . . . . 6 |- ( G e. V -> G e. _V )
4	3	adantr 276	. . . . 5 |- ( ( G e. V /\ S e. W ) -> G e. _V )
5		elex 2750	. . . . . 6 |- ( S e. W -> S e. _V )
6	5	adantl 277	. . . . 5 |- ( ( G e. V /\ S e. W ) -> S e. _V )
7		vex 2742	. . . . . . . . 9 |- x e. _V
8		vex 2742	. . . . . . . . 9 |- y e. _V
9	7, 8	prss 3750	. . . . . . . 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 4072	. . . . . 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 12522	. . . . . . . . 9 |- Base Fn _V
13		funfvex 5534	. . . . . . . . . 10 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
14	13	funfni 5318	. . . . . . . . 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 4742	. . . . . . . 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 4702	. . . . . . . 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 4145	. . . . . 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 2265	. . . . 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 5517	. . . . . . . . 9 |- ( r = G -> ( Base ` r ) = ( Base ` G ) )
23	22	sseq2d 3187	. . . . . . . 8 |- ( r = G -> ( { x , y } C_ ( Base ` r ) <-> { x , y } C_ ( Base ` G ) ) )
24		fveq2 5517	. . . . . . . . . 10 |- ( r = G -> ( +g ` r ) = ( +g ` G ) )
25		fveq2 5517	. . . . . . . . . . 11 |- ( r = G -> ( invg ` r ) = ( invg ` G ) )
26	25	fveq1d 5519	. . . . . . . . . 10 |- ( r = G -> ( ( invg ` r ) ` x ) = ( ( invg ` G ) ` x ) )
27		eqidd 2178	. . . . . . . . . 10 |- ( r = G -> y = y )
28	24, 26, 27	oveq123d 5898	. . . . . . . . 9 |- ( r = G -> ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) = ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) )
29	28	eleq1d 2246	. . . . . . . 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 4071	. . . . . 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 2241	. . . . . . . 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 4071	. . . . . 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 13037	. . . . . 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 6011	. . . . 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 1238	. . . 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 2222	. . 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 4712	. 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 1353   e. wcel 2148   _Vcvv 2739   C_ wss 3131   {cpr 3595   {copab 4065   X. cxp 4626   Rel wrel 4633   Fn wfn 5213   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   invgcminusg 12883   ~_QG cqg 13034

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-eqg 13037

This theorem is referenced by:  eqger  13088  eqgid  13090