Theorem psmetutop 20156

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 20145	. . . . . . . . . . . 12 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (metUnif ` D ) e. (UnifOn ` X ) )
2		utopval 19805	. . . . . . . . . . . 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 2508	. . . . . . . . . 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 2895	. . . . . . . . . 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 3868	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. (unifTop ` (metUnif ` D ) ) ) -> a C_ X )
10		unirnblps 19992	. . . . . . 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 3391	. . . . 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 3355	. . . . . . . . 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 20154	. . . . . . . . 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 1218	. . . . . . . 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 3362	. . . . . . . . . . 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 2825	. . . . . . . 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 2812	. . . . . . 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 2860	. . . . . 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 2797	. . . . 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 5699	. . . . . . 7 |- ( ball ` D ) e. _V
32	31	rnex 6510	. . . . . 6 |- ran ( ball ` D ) e. _V
33		eltg2 18561	. . . . . 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 3390	. . . . . 6 |- ( ( ( X =/= (/) /\ D e. (PsMet ` X ) ) /\ a e. ( topGen ` ran ( ball ` D ) ) ) -> a C_ X )
42		elpwg 3866	. . . . . . 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 3354	. . . . . . . . 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 2812	. . . . . . . . . 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 19999	. . . . . . . . . . 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 20148	. . . . . . . . . . . . 13 |- ( ( D e. (PsMet ` X ) /\ x e. X /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
53	52	3expa 1187	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ x e. X ) /\ d e. RR+ ) -> ( x ( ball ` D ) d ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
54	53	sseq1d 3381	. . . . . . . . . . 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 2730	. . . . . . . . . 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 1217	. . . . . . . 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 6522	. . . . . . . . . . 11 |- ( D e. (PsMet ` X ) -> `' D e. _V )
59		imaexg 6513	. . . . . . . . . . 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 2798	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> A. d e. RR+ ( `' D " ( 0 [,) d ) ) e. _V )
62		eqid 2441	. . . . . . . . . 10 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) )
63		imaeq1 5162	. . . . . . . . . . 11 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( v " { x } ) = ( ( `' D " ( 0 [,) d ) ) " { x } ) )
64	63	sseq1d 3381	. . . . . . . . . 10 |- ( v = ( `' D " ( 0 [,) d ) ) -> ( ( v " { x } ) C_ a <-> ( ( `' D " ( 0 [,) d ) ) " { x } ) C_ a ) )
65	62, 64	rexrnmpt 5851	. . . . . . . . 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 6097	. . . . . . . . . . . . . . 15 |- ( d = e -> ( 0 [,) d ) = ( 0 [,) e ) )
69	68	imaeq2d 5167	. . . . . . . . . . . . . 14 |- ( d = e -> ( `' D " ( 0 [,) d ) ) = ( `' D " ( 0 [,) e ) ) )
70	69	cbvmptv 4381	. . . . . . . . . . . . 13 |- ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
71	70	rneqi 5064	. . . . . . . . . . . 12 |- ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) = ran ( e e. RR+ |-> ( `' D " ( 0 [,) e ) ) )
72	71	metustfbas 20139	. . . . . . . . . . 11 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) e. ( fBas ` ( X X. X ) ) )
73		ssfg 19443	. . . . . . . . . . 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 20123	. . . . . . . . . . 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 3391	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ran ( d e. RR+ |-> ( `' D " ( 0 [,) d ) ) ) C_ (metUnif ` D ) )
78		ssrexv 3415	. . . . . . . . 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 2797	. . . . 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 828	. 2 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( a e. (unifTop ` (metUnif ` D ) ) <-> a e. ( topGen ` ran ( ball ` D ) ) ) )
87	86	eqrdv 2439	1 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> (unifTop ` (metUnif ` D ) ) = ( topGen ` ran ( ball ` D ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2604   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970   C_ wss 3326   (/)c0 3635   ~Pcpw 3858   {csn 3875   U.cuni 4089   e. cmpt 4348   X. cxp 4836   `'ccnv 4837   ran crn 4839   "cima 4841   ` cfv 5416  (class class class)co 6089   0cc0 9280   RR+crp 10989   [,)cico 11300   topGenctg 14374  PsMetcpsmet 17798   ballcbl 17801   fBascfbas 17802   filGencfg 17803  metUnifcmetu 17806  UnifOncust 19772  unifTopcutop 19803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-sup 7689  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-q 10952  df-rp 10990  df-xneg 11087  df-xadd 11088  df-xmul 11089  df-ico 11304  df-topgen 14380  df-psmet 17807  df-bl 17810  df-fbas 17812  df-fg 17813  df-metu 17815  df-fil 19417  df-ust 19773  df-utop 19804

This theorem is referenced by:  xmetutop  20157