Theorem psmetutop 22372

Description: The topology induced by a uniform structure generated by a metric D is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)

Assertion

Ref	Expression
psmetutop	|- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( topGen ` ran ( ball ` D ) ) )

Proof of Theorem psmetutop

Dummy variables a b d e v x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		metuust 22365	. . . . . . . . . . . 12 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) e. (UnifOn ` X ) )
2		utopval 22036	. . . . . . . . . . . 12 |- ( (metUnif ` D ) e. (UnifOn ` X ) -> (unifTop ` (metUnif ` D ) ) = { a e. ~P X | A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a } )
3	1, 2	syl 17	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = { a e. ~P X | A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a } )
4	3	eleq2d 2687	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( a e. (unifTop ` (metUnif ` D ) ) <-> a e. { a e. ~P X | A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a } ) )
5		rabid 3116	. . . . . . . . . 10 |- ( a e. { a e. ~P X | A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a } <-> ( a e. ~P X /\ A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a ) )
6	4, 5	syl6bb 276	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( a e. (unifTop ` (metUnif ` D ) ) <-> ( a e. ~P X /\ A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a ) ) )
7	6	biimpa 501	. . . . . . . 8 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> ( a e. ~P X /\ A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a ) )
8	7	simpld 475	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a e. ~P X )
9	8	elpwid 4170	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ X )
10		unirnblps 22224	. . . . . . 7 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )
11	10	ad2antlr 763	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> U. ran ( ball ` D ) = X )
12	9, 11	sseqtr4d 3642	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ U. ran ( ball ` D ) )
13		simpr 477	. . . . . . . 8 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) /\ v e. (metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> ( v " { x } ) C_ a )
14		simp-5r 809	. . . . . . . . 9 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) /\ v e. (metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> D e. (PsMet ` X ) )
15		simplr 792	. . . . . . . . 9 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) /\ v e. (metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> v e. (metUnif ` D ) )
16	9	ad3antrrr 766	. . . . . . . . . 10 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) /\ v e. (metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> a C_ X )
17		simpllr 799	. . . . . . . . . 10 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) /\ v e. (metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> x e. a )
18	16, 17	sseldd 3604	. . . . . . . . 9 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) /\ v e. (metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> x e. X )
19		metustbl 22371	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ v e. (metUnif ` D ) /\ x e. X ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ ( v " { x } ) ) )
20	14, 15, 18, 19	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) /\ v e. (metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ ( v " { x } ) ) )
21		sstr 3611	. . . . . . . . . . 11 |- ( ( b C_ ( v " { x } ) /\ ( v " { x } ) C_ a ) -> b C_ a )
22	21	expcom 451	. . . . . . . . . 10 |- ( ( v " { x } ) C_ a -> ( b C_ ( v " { x } ) -> b C_ a ) )
23	22	anim2d 589	. . . . . . . . 9 |- ( ( v " { x } ) C_ a -> ( ( x e. b /\ b C_ ( v " { x } ) ) -> ( x e. b /\ b C_ a ) ) )
24	23	reximdv 3016	. . . . . . . 8 |- ( ( v " { x } ) C_ a -> ( E. b e. ran ( ball ` D ) ( x e. b /\ b C_ ( v " { x } ) ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) )
25	13, 20, 24	sylc 65	. . . . . . 7 |- ( ( ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) /\ v e. (metUnif ` D ) ) /\ ( v " { x } ) C_ a ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) )
26	7	simprd 479	. . . . . . . 8 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a )
27	26	r19.21bi 2932	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) -> E. v e. (metUnif ` D ) ( v " { x } ) C_ a )
28	25, 27	r19.29a 3078	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) /\ x e. a ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) )
29	28	ralrimiva 2966	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) )
30	12, 29	jca 554	. . . 4 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) )
31		fvex 6201	. . . . . 6 |- ( ball ` D ) e. _V
32	31	rnex 7100	. . . . 5 |- ran ( ball ` D ) e. _V
33		eltg2 20762	. . . . 5 |- ( ran ( ball ` D ) e. _V -> ( a e. ( topGen ` ran ( ball ` D ) ) <-> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) )
34	32, 33	mp1i 13	. . . 4 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> ( a e. ( topGen ` ran ( ball ` D ) ) <-> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) )
35	30, 34	mpbird 247	. . 3 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a e. ( topGen ` ran ( ball ` D ) ) )
36	32, 33	mp1i 13	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( a e. ( topGen ` ran ( ball ` D ) ) <-> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) ) )
37	36	biimpa 501	. . . . . . . 8 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a C_ U. ran ( ball ` D ) /\ A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) ) )
38	37	simpld 475	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ U. ran ( ball ` D ) )
39	10	ad2antlr 763	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> U. ran ( ball ` D ) = X )
40	38, 39	sseqtrd 3641	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ X )
41		elpwg 4166	. . . . . . 7 |- ( a e. ( topGen ` ran ( ball ` D ) ) -> ( a e. ~P X <-> a C_ X ) )
42	41	adantl 482	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a e. ~P X <-> a C_ X ) )
43	40, 42	mpbird 247	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. ~P X )
44		simpllr 799	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> D e. (PsMet ` X ) )
45	40	sselda 3603	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> x e. X )
46	37	simprd 479	. . . . . . . . . . 11 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> A. x e. a E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) )
47	46	r19.21bi 2932	. . . . . . . . . 10 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) )
48		blssexps 22231	. . . . . . . . . . 11 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) <-> E. d e. RR+ ( x ( ball ` D ) d ) C_ a ) )
49	44, 45, 48	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> ( E. b e. ran ( ball ` D ) ( x e. b /\ b C_ a ) <-> E. d e. RR+ ( x ( ball ` D ) d ) C_ a ) )
50	47, 49	mpbid 222	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. d e. RR+ ( x ( ball ` D ) d ) C_ a )
51		blval2 22367	. . . . . . . . . . . . 13 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
52	51	3expa 1265	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
53	52	sseq1d 3632	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( ( x ( ball ` D ) d ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
54	53	rexbidva 3049	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ x e. X ) -> ( E. d e. RR+ ( x ( ball ` D ) d ) C_ a <-> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
55	54	biimpa 501	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ x e. X ) /\ E. d e. RR+ ( x ( ball ` D ) d ) C_ a ) -> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a )
56	44, 45, 50, 55	syl21anc 1325	. . . . . . . 8 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a )
57		cnvexg 7112	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> `' D e. _V )
58		imaexg 7103	. . . . . . . . . . 11 |- ( `' D e. _V -> ( `' D " ( 0 [,) d ) ) e. _V )
59	57, 58	syl 17	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> ( `' D " ( 0 [,) d ) ) e. _V )
60	59	ralrimivw 2967	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> A. d e. RR+ ( `' D " ( 0 [,) d ) ) e. _V )
61		eqid 2622	. . . . . . . . . 10 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
62		imaeq1 5461	. . . . . . . . . . 11 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( v " { x } ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
63	62	sseq1d 3632	. . . . . . . . . 10 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( ( v " { x } ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
64	61, 63	rexrnmpt 6369	. . . . . . . . 9 |- ( A. d e. RR+ ( `' D " ( 0 [,) d ) ) e. _V -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a <-> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
65	44, 60, 64	3syl 18	. . . . . . . 8 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a <-> E. d e. RR+ ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
66	56, 65	mpbird 247	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a )
67		oveq2 6658	. . . . . . . . . . . . . . 15 |- ( d = e -> ( 0 [,) d ) = ( 0 [,) e ) )
68	67	imaeq2d 5466	. . . . . . . . . . . . . 14 |- ( d = e -> ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) e ) ) )
69	68	cbvmptv 4750	. . . . . . . . . . . . 13 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
70	69	rneqi 5352	. . . . . . . . . . . 12 |- ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ran ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
71	70	metustfbas 22362	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) e. ( fBas ` ( X X. X ) ) )
72		ssfg 21676	. . . . . . . . . . 11 |- ( ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) e. ( fBas ` ( X X. X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
73	71, 72	syl 17	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
74		metuval 22354	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> (metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
75	74	adantl 482	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
76	73, 75	sseqtr4d 3642	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ (metUnif ` D ) )
77		ssrexv 3667	. . . . . . . . 9 |- ( ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ (metUnif ` D ) -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a -> E. v e. (metUnif ` D ) ( v " { x } ) C_ a ) )
78	76, 77	syl 17	. . . . . . . 8 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a -> E. v e. (metUnif ` D ) ( v " { x } ) C_ a ) )
79	78	ad2antrr 762	. . . . . . 7 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> ( E. v e. ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ( v " { x } ) C_ a -> E. v e. (metUnif ` D ) ( v " { x } ) C_ a ) )
80	66, 79	mpd 15	. . . . . 6 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> E. v e. (metUnif ` D ) ( v " { x } ) C_ a )
81	80	ralrimiva 2966	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a )
82	43, 81	jca 554	. . . 4 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a e. ~P X /\ A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a ) )
83	6	biimpar 502	. . . 4 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ ( a e. ~P X /\ A. x e. a E. v e. (metUnif ` D ) ( v " { x } ) C_ a ) ) -> a e. (unifTop ` (metUnif ` D ) ) )
84	82, 83	syldan 487	. . 3 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. (unifTop ` (metUnif ` D ) ) )
85	35, 84	impbida 877	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( a e. (unifTop ` (metUnif ` D ) ) <-> a e. ( topGen ` ran ( ball ` D ) ) ) )
86	85	eqrdv 2620	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( topGen ` ran ( ball ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR+crp 11832   [,)cico 12177   topGenctg 16098  PsMetcpsmet 19730   ballcbl 19733   fBascfbas 19734   filGencfg 19735  metUnifcmetu 19737  UnifOncust 22003  unifTopcutop 22034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-topgen 16104  df-psmet 19738  df-bl 19741  df-fbas 19743  df-fg 19744  df-metu 19745  df-fil 21650  df-ust 22004  df-utop 22035

This theorem is referenced by:  xmetutop  22373