Theorem psmetutop 21660

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 21653	. . . . . . . . . . . 12 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) e. (UnifOn ` X ) )
2		utopval 21325	. . . . . . . . . . . 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 2534	. . . . . . . . . 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 2953	. . . . . . . . . 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 269	. . . . . . . . 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 492	. . . . . . . 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 466	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a e. ~P X )
9	8	elpwid 3952	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ X )
10		unirnblps 21512	. . . . . . 7 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )
11	10	ad2antlr 741	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> U. ran ( ball ` D ) = X )
12	9, 11	sseqtr4d 3455	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ U. ran ( ball ` D ) )
13		simpr 468	. . . . . . . 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 787	. . . . . . . . 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 770	. . . . . . . . 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 744	. . . . . . . . . 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 777	. . . . . . . . . 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 3419	. . . . . . . . 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 21659	. . . . . . . . 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 1292	. . . . . . . 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 3426	. . . . . . . . . . 11 |- ( ( b C_ ( v " { x } ) /\ ( v " { x } ) C_ a ) -> b C_ a )
22	21	expcom 442	. . . . . . . . . 10 |- ( ( v " { x } ) C_ a -> ( b C_ ( v " { x } ) -> b C_ a ) )
23	22	anim2d 575	. . . . . . . . 9 |- ( ( v " { x } ) C_ a -> ( ( x e. b /\ b C_ ( v " { x } ) ) -> ( x e. b /\ b C_ a ) ) )
24	23	reximdv 2857	. . . . . . . 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 61	. . . . . . 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 470	. . . . . . . 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 2776	. . . . . . 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 2918	. . . . . 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 2809	. . . . 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 541	. . . 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 5889	. . . . . 6 |- ( ball ` D ) e. _V
32	31	rnex 6746	. . . . 5 |- ran ( ball ` D ) e. _V
33		eltg2 20050	. . . . 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 240	. . 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 492	. . . . . . . 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 466	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ U. ran ( ball ` D ) )
39	10	ad2antlr 741	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> U. ran ( ball ` D ) = X )
40	38, 39	sseqtrd 3454	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ X )
41		elpwg 3950	. . . . . . 7 |- ( a e. ( topGen ` ran ( ball ` D ) ) -> ( a e. ~P X <-> a C_ X ) )
42	41	adantl 473	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a e. ~P X <-> a C_ X ) )
43	40, 42	mpbird 240	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. ~P X )
44		simpllr 777	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> D e. (PsMet ` X ) )
45	40	sselda 3418	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> x e. X )
46	37	simprd 470	. . . . . . . . . . 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 2776	. . . . . . . . . 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 21519	. . . . . . . . . . 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 673	. . . . . . . . . 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 215	. . . . . . . . 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 21655	. . . . . . . . . . . . 13 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
52	51	3expa 1231	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
53	52	sseq1d 3445	. . . . . . . . . . 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 2889	. . . . . . . . . 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 492	. . . . . . . . 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 1291	. . . . . . . 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 6758	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> `' D e. _V )
58		imaexg 6749	. . . . . . . . . . 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 2810	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> A. d e. RR+ ( `' D " ( 0 [,) d ) ) e. _V )
61		eqid 2471	. . . . . . . . . 10 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
62		imaeq1 5169	. . . . . . . . . . 11 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( v " { x } ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
63	62	sseq1d 3445	. . . . . . . . . 10 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( ( v " { x } ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
64	61, 63	rexrnmpt 6047	. . . . . . . . 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 240	. . . . . . 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 6316	. . . . . . . . . . . . . . 15 |- ( d = e -> ( 0 [,) d ) = ( 0 [,) e ) )
68	67	imaeq2d 5174	. . . . . . . . . . . . . 14 |- ( d = e -> ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) e ) ) )
69	68	cbvmptv 4488	. . . . . . . . . . . . 13 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
70	69	rneqi 5067	. . . . . . . . . . . 12 |- ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ran ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
71	70	metustfbas 21650	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) e. ( fBas ` ( X X. X ) ) )
72		ssfg 20965	. . . . . . . . . . 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 21642	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> (metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
75	74	adantl 473	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
76	73, 75	sseqtr4d 3455	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ (metUnif ` D ) )
77		ssrexv 3480	. . . . . . . . 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 740	. . . . . . 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 2809	. . . . 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 541	. . . 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 493	. . . 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 478	. . 3 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. (unifTop ` (metUnif ` D ) ) )
85	35, 84	impbida 850	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( a e. (unifTop ` (metUnif ` D ) ) <-> a e. ( topGen ` ran ( ball ` D ) ) ) )
86	85	eqrdv 2469	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( topGen ` ran ( ball ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   {csn 3959   U.cuni 4190   |-> cmpt 4454   X. cxp 4837   `'ccnv 4838   ran crn 4840   "cima 4842   ` cfv 5589  (class class class)co 6308   0cc0 9557   RR+crp 11325   [,)cico 11662   topGenctg 15414  PsMetcpsmet 19031   ballcbl 19034   fBascfbas 19035   filGencfg 19036  metUnifcmetu 19038  UnifOncust 21292  unifTopcutop 21323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ico 11666  df-topgen 15420  df-psmet 19039  df-bl 19042  df-fbas 19044  df-fg 19045  df-metu 19046  df-fil 20939  df-ust 21293  df-utop 21324

This theorem is referenced by:  xmetutop  21661