Theorem psmetutop 20814

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 20803	. . . . . . . . . . . 12 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) e. (UnifOn ` X ) )
2		utopval 20463	. . . . . . . . . . . 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 2530	. . . . . . . . . 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 484	. . . . . . . 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 459	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a e. ~P X )
9	8	elpwid 4013	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ X )
10		unirnblps 20650	. . . . . . 7 |- ( D e. (PsMet ` X ) -> U. ran ( ball ` D ) = X )
11	10	ad2antlr 726	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> U. ran ( ball ` D ) = X )
12	9, 11	sseqtr4d 3534	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ U. ran ( ball ` D ) )
13		simpr 461	. . . . . . . 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 754	. . . . . . . . 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 729	. . . . . . . . . 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 3498	. . . . . . . . 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 20812	. . . . . . . . 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 1223	. . . . . . . 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 3505	. . . . . . . . . . 11 |- ( ( b C_ ( v " { x } ) /\ ( v " { x } ) C_ a ) -> b C_ a )
22	21	expcom 435	. . . . . . . . . 10 |- ( ( v " { x } ) C_ a -> ( b C_ ( v " { x } ) -> b C_ a ) )
23	22	anim2d 565	. . . . . . . . 9 |- ( ( v " { x } ) C_ a -> ( ( x e. b /\ b C_ ( v " { x } ) ) -> ( x e. b /\ b C_ a ) ) )
24	23	reximdv 2930	. . . . . . . 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 463	. . . . . . . 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 2826	. . . . . . 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 2871	. . . . 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 532	. . . 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 5867	. . . . . . 7 |- ( ball ` D ) e. _V
32	31	rnex 6708	. . . . . 6 |- ran ( ball ` D ) e. _V
33		eltg2 19219	. . . . . 6 |- ( 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	ax-mp 5	. . . . 5 |- ( 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	34	a1i 11	. . . 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 ) ) ) )
36	30, 35	mpbird 232	. . 3 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a e. ( topGen ` ran ( ball ` D ) ) )
37	34	a1i 11	. . . . . . . . 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 ) ) ) )
38	37	biimpa 484	. . . . . . . 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 ) ) )
39	38	simpld 459	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ U. ran ( ball ` D ) )
40	10	ad2antlr 726	. . . . . . 7 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> U. ran ( ball ` D ) = X )
41	39, 40	sseqtrd 3533	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ X )
42		elpwg 4011	. . . . . . 7 |- ( a e. ( topGen ` ran ( ball ` D ) ) -> ( a e. ~P X <-> a C_ X ) )
43	42	adantl 466	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> ( a e. ~P X <-> a C_ X ) )
44	41, 43	mpbird 232	. . . . 5 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. ~P X )
45		simpllr 758	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> D e. (PsMet ` X ) )
46	41	sselda 3497	. . . . . . . . 9 |- ( ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) /\ x e. a ) -> x e. X )
47	38	simprd 463	. . . . . . . . . . 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 ) )
48	47	r19.21bi 2826	. . . . . . . . . 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 ) )
49		blssexps 20657	. . . . . . . . . . 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 ) )
50	45, 46, 49	syl2anc 661	. . . . . . . . . 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 ) )
51	48, 50	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 )
52		blval2 20806	. . . . . . . . . . . . 13 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
53	52	3expa 1191	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
54	53	sseq1d 3524	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( ( x ( ball ` D ) d ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
55	54	rexbidva 2963	. . . . . . . . . 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 ) )
56	55	biimpa 484	. . . . . . . . 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 )
57	45, 46, 51, 56	syl21anc 1222	. . . . . . . 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 )
58		cnvexg 6720	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> `' D e. _V )
59		imaexg 6711	. . . . . . . . . . 11 |- ( `' D e. _V -> ( `' D " ( 0 [,) d ) ) e. _V )
60	58, 59	syl 16	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> ( `' D " ( 0 [,) d ) ) e. _V )
61	60	ralrimivw 2872	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> A. d e. RR+ ( `' D " ( 0 [,) d ) ) e. _V )
62		eqid 2460	. . . . . . . . . 10 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
63		imaeq1 5323	. . . . . . . . . . 11 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( v " { x } ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
64	63	sseq1d 3524	. . . . . . . . . 10 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( ( v " { x } ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
65	62, 64	rexrnmpt 6022	. . . . . . . . 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 ) )
66	45, 61, 65	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 ) )
67	57, 66	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 )
68		oveq2 6283	. . . . . . . . . . . . . . 15 |- ( d = e -> ( 0 [,) d ) = ( 0 [,) e ) )
69	68	imaeq2d 5328	. . . . . . . . . . . . . 14 |- ( d = e -> ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) e ) ) )
70	69	cbvmptv 4531	. . . . . . . . . . . . 13 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
71	70	rneqi 5220	. . . . . . . . . . . 12 |- ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ran ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
72	71	metustfbas 20797	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) e. ( fBas ` ( X X. X ) ) )
73		ssfg 20101	. . . . . . . . . . 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 ) ) ) ) )
74	72, 73	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 ) ) ) ) )
75		metuval 20781	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> (metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
76	75	adantl 466	. . . . . . . . . 10 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) = ( ( X X. X ) filGen ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) ) )
77	74, 76	sseqtr4d 3534	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ (metUnif ` D ) )
78		ssrexv 3558	. . . . . . . . 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 ) )
79	77, 78	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 ) )
80	79	ad2antrr 725	. . . . . . 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 ) )
81	67, 80	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 )
82	81	ralrimiva 2871	. . . . 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 )
83	44, 82	jca 532	. . . 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 ) )
84	6	biimpar 485	. . . 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 ) ) )
85	83, 84	syldan 470	. . 3 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a e. (unifTop ` (metUnif ` D ) ) )
86	36, 85	impbida 829	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( a e. (unifTop ` (metUnif ` D ) ) <-> a e. ( topGen ` ran ( ball ` D ) ) ) )
87	86	eqrdv 2457	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( topGen ` ran ( ball ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1374   e. wcel 1762   =/= wne 2655   A.wral 2807   E.wrex 2808   {crab 2811   _Vcvv 3106   C_ wss 3469   (/)c0 3778   ~Pcpw 4003   {csn 4020   U.cuni 4238   |-> cmpt 4498   X. cxp 4990   `'ccnv 4991   ran crn 4993   "cima 4995   ` cfv 5579  (class class class)co 6275   0cc0 9481   RR+crp 11209   [,)cico 11520   topGenctg 14682  PsMetcpsmet 18166   ballcbl 18169   fBascfbas 18170   filGencfg 18171  metUnifcmetu 18174  UnifOncust 20430  unifTopcutop 20461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-sup 7890  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-n0 10785  df-z 10854  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ico 11524  df-topgen 14688  df-psmet 18175  df-bl 18178  df-fbas 18180  df-fg 18181  df-metu 18183  df-fil 20075  df-ust 20431  df-utop 20462

This theorem is referenced by:  xmetutop  20815