Theorem releqgg 13806

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 4856	. 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 2814	. . . . . 6 |- ( G e. V -> G e. _V )
4	3	adantr 276	. . . . 5 |- ( ( G e. V /\ S e. W ) -> G e. _V )
5		elex 2814	. . . . . 6 |- ( S e. W -> S e. _V )
6	5	adantl 277	. . . . 5 |- ( ( G e. V /\ S e. W ) -> S e. _V )
7		vex 2805	. . . . . . . . 9 |- x e. _V
8		vex 2805	. . . . . . . . 9 |- y e. _V
9	7, 8	prss 3829	. . . . . . . 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 4156	. . . . . 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 13140	. . . . . . . . 9 |- Base Fn _V
13		funfvex 5656	. . . . . . . . . 10 |- ( ( Fun Base /\ G e. dom Base ) -> ( Base ` G ) e. _V )
14	13	funfni 5432	. . . . . . . . 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 4840	. . . . . . . 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 4800	. . . . . . . 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 4229	. . . . . 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 2319	. . . . 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 5639	. . . . . . . . 9 |- ( r = G -> ( Base ` r ) = ( Base ` G ) )
23	22	sseq2d 3257	. . . . . . . 8 |- ( r = G -> ( { x , y } C_ ( Base ` r ) <-> { x , y } C_ ( Base ` G ) ) )
24		fveq2 5639	. . . . . . . . . 10 |- ( r = G -> ( +g ` r ) = ( +g ` G ) )
25		fveq2 5639	. . . . . . . . . . 11 |- ( r = G -> ( invg ` r ) = ( invg ` G ) )
26	25	fveq1d 5641	. . . . . . . . . 10 |- ( r = G -> ( ( invg ` r ) ` x ) = ( ( invg ` G ) ` x ) )
27		eqidd 2232	. . . . . . . . . 10 |- ( r = G -> y = y )
28	24, 26, 27	oveq123d 6038	. . . . . . . . 9 |- ( r = G -> ( ( ( invg ` r ) ` x ) ( +g ` r ) y ) = ( ( ( invg ` G ) ` x ) ( +g ` G ) y ) )
29	28	eleq1d 2300	. . . . . . . 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 4155	. . . . . 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 2295	. . . . . . . 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 4155	. . . . . 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 13758	. . . . . 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 6155	. . . . 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 1273	. . . 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 2276	. . 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 4810	. 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 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   {cpr 3670   {copab 4149   X. cxp 4723   Rel wrel 4730   Fn wfn 5321   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   invgcminusg 13583   ~_QG cqg 13755

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-inn 9143  df-ndx 13084  df-slot 13085  df-base 13087  df-eqg 13758

This theorem is referenced by:  eqger  13810  eqgid  13812