Theorem psmetutop 21255

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 21244	. . . . . . . . . . . 12 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) e. (UnifOn ` X ) )
2		utopval 20904	. . . . . . . . . . . 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 16	. . . . . . . . . . 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 2524	. . . . . . . . . 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 3031	. . . . . . . . . 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 261	. . . . . . . . 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 482	. . . . . . . 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 457	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a e. ~P X )
9	8	elpwid 4009	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ X )
10		unirnblps 21091	. . . . . . 7 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )
11	10	ad2antlr 724	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> U. ran ( ball ` D ) = X )
12	9, 11	sseqtr4d 3526	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ U. ran ( ball ` D ) )
13		simpr 459	. . . . . . . 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 768	. . . . . . . . 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 753	. . . . . . . . 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 727	. . . . . . . . . 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 758	. . . . . . . . . 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 3490	. . . . . . . . 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 21253	. . . . . . . . 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 1226	. . . . . . . 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 3497	. . . . . . . . . . 11 |- ( ( b C_ ( v " { x } ) /\ ( v " { x } ) C_ a ) -> b C_ a )
22	21	expcom 433	. . . . . . . . . 10 |- ( ( v " { x } ) C_ a -> ( b C_ ( v " { x } ) -> b C_ a ) )
23	22	anim2d 563	. . . . . . . . 9 |- ( ( v " { x } ) C_ a -> ( ( x e. b /\ b C_ ( v " { x } ) ) -> ( x e. b /\ b C_ a ) ) )
24	23	reximdv 2928	. . . . . . . 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 60	. . . . . . 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 461	. . . . . . . 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 2823	. . . . . . 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 2996	. . . . . 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 2868	. . . . 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 530	. . . 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 5858	. . . . . 6 |- ( ball ` D ) e. _V
32	31	rnex 6707	. . . . 5 |- ran ( ball ` D ) e. _V
33		eltg2 19629	. . . . 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 12	. . . 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 232	. . 3 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a e. ( topGen ` ran ( ball ` D ) ) )
36	32, 33	mp1i 12	. . . . . . . . 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 482	. . . . . . . 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 457	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ U. ran ( ball ` D ) )
39	10	ad2antlr 724	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> U. ran ( ball ` D ) = X )
40	38, 39	sseqtrd 3525	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ X )
41		elpwg 4007	. . . . . . 7 |- ( a e. ( topGen ` ran ( ball ` D ) ) -> ( a e. ~P X <-> a C_ X ) )
42	41	adantl 464	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a e. ~P X <-> a C_ X ) )
43	40, 42	mpbird 232	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. ~P X )
44		simpllr 758	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> D e. (PsMet ` X ) )
45	40	sselda 3489	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> x e. X )
46	37	simprd 461	. . . . . . . . . . 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 2823	. . . . . . . . . 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 21098	. . . . . . . . . . 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 659	. . . . . . . . . 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 210	. . . . . . . . 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 21247	. . . . . . . . . . . . 13 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
52	51	3expa 1194	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
53	52	sseq1d 3516	. . . . . . . . . . 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 2962	. . . . . . . . . 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 482	. . . . . . . . 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 1225	. . . . . . . 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 6719	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> `' D e. _V )
58		imaexg 6710	. . . . . . . . . . 11 |- ( `' D e. _V -> ( `' D " ( 0 [,) d ) ) e. _V )
59	57, 58	syl 16	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> ( `' D " ( 0 [,) d ) ) e. _V )
60	59	ralrimivw 2869	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> A. d e. RR+ ( `' D " ( 0 [,) d ) ) e. _V )
61		eqid 2454	. . . . . . . . . 10 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
62		imaeq1 5320	. . . . . . . . . . 11 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( v " { x } ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
63	62	sseq1d 3516	. . . . . . . . . 10 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( ( v " { x } ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
64	61, 63	rexrnmpt 6017	. . . . . . . . 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 20	. . . . . . . 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 232	. . . . . . 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 6278	. . . . . . . . . . . . . . 15 |- ( d = e -> ( 0 [,) d ) = ( 0 [,) e ) )
68	67	imaeq2d 5325	. . . . . . . . . . . . . 14 |- ( d = e -> ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) e ) ) )
69	68	cbvmptv 4530	. . . . . . . . . . . . 13 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
70	69	rneqi 5218	. . . . . . . . . . . 12 |- ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ran ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
71	70	metustfbas 21238	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) e. ( fBas ` ( X X. X ) ) )
72		ssfg 20542	. . . . . . . . . . 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 16	. . . . . . . . . 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 21222	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> (metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
75	74	adantl 464	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
76	73, 75	sseqtr4d 3526	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ (metUnif ` D ) )
77		ssrexv 3551	. . . . . . . . 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 16	. . . . . . . 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 723	. . . . . . 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 2868	. . . . 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 530	. . . 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 483	. . . 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 468	. . 3 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. (unifTop ` (metUnif ` D ) ) )
85	35, 84	impbida 830	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( a e. (unifTop ` (metUnif ` D ) ) <-> a e. ( topGen ` ran ( ball ` D ) ) ) )
86	85	eqrdv 2451	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( topGen ` ran ( ball ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   C_ wss 3461   (/)c0 3783   ~Pcpw 3999   {csn 4016   U.cuni 4235   |-> cmpt 4497   X. cxp 4986   `'ccnv 4987   ran crn 4989   "cima 4991   ` cfv 5570  (class class class)co 6270   0cc0 9481   RR+crp 11221   [,)cico 11534   topGenctg 14930  PsMetcpsmet 18600   ballcbl 18603   fBascfbas 18604   filGencfg 18605  metUnifcmetu 18608  UnifOncust 20871  unifTopcutop 20902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-topgen 14936  df-psmet 18609  df-bl 18612  df-fbas 18614  df-fg 18615  df-metu 18617  df-fil 20516  df-ust 20872  df-utop 20903

This theorem is referenced by:  xmetutop  21256