Theorem restmetu 21663

Description: The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)

Assertion

Ref	Expression
restmetu	|- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( (metUnif ` D ) ↾t ( A X. A ) ) = (metUnif ` ( D |` ( A X. A ) ) ) )

Proof of Theorem restmetu

Dummy variables a b u v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1030	. . . 4 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> A =/= (/) )
2		psmetres2 21408	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ A C_ X ) -> ( D |` ( A X. A ) ) e. (PsMet ` A ) )
3	2	3adant1 1048	. . . 4 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( D |` ( A X. A ) ) e. (PsMet ` A ) )
4		oveq2 6316	. . . . . . . 8 |- ( a = b -> ( 0 [,) a ) = ( 0 [,) b ) )
5	4	imaeq2d 5174	. . . . . . 7 |- ( a = b -> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) )
6	5	cbvmptv 4488	. . . . . 6 |- ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) = ( b e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) )
7	6	rneqi 5067	. . . . 5 |- ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) = ran ( b e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) )
8	7	metustfbas 21650	. . . 4 |- ( ( A =/= (/) /\ ( D |` ( A X. A ) ) e. (PsMet ` A ) ) -> ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) e. ( fBas ` ( A X. A ) ) )
9	1, 3, 8	syl2anc 673	. . 3 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) e. ( fBas ` ( A X. A ) ) )
10		fgval 20963	. . 3 |- ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) e. ( fBas ` ( A X. A ) ) -> ( ( A X. A ) filGen ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) ) = { v e. ~P ( A X. A ) | ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P v ) =/= (/) } )
11	9, 10	syl 17	. 2 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( ( A X. A ) filGen ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) ) = { v e. ~P ( A X. A ) | ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P v ) =/= (/) } )
12		metuval 21642	. . 3 |- ( ( D |` ( A X. A ) ) e. (PsMet ` A ) -> (metUnif ` ( D |` ( A X. A ) ) ) = ( ( A X. A ) filGen ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) ) )
13	3, 12	syl 17	. 2 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> (metUnif ` ( D |` ( A X. A ) ) ) = ( ( A X. A ) filGen ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) ) )
14		fvex 5889	. . . 4 |- (metUnif ` D ) e. _V
15	3	elfvexd 5907	. . . . 5 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> A e. _V )
16		xpexg 6612	. . . . 5 |- ( ( A e. _V /\ A e. _V ) -> ( A X. A ) e. _V )
17	15, 15, 16	syl2anc 673	. . . 4 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( A X. A ) e. _V )
18		restval 15403	. . . 4 |- ( ( (metUnif ` D ) e. _V /\ ( A X. A ) e. _V ) -> ( (metUnif ` D ) ↾t ( A X. A ) ) = ran ( v e. (metUnif ` D ) |-> ( v i^i ( A X. A ) ) ) )
19	14, 17, 18	sylancr 676	. . 3 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( (metUnif ` D ) ↾t ( A X. A ) ) = ran ( v e. (metUnif ` D ) |-> ( v i^i ( A X. A ) ) ) )
20		inss2 3644	. . . . . . . . . . 11 |- ( v i^i ( A X. A ) ) C_ ( A X. A )
21		sseq1 3439	. . . . . . . . . . 11 |- ( u = ( v i^i ( A X. A ) ) -> ( u C_ ( A X. A ) <-> ( v i^i ( A X. A ) ) C_ ( A X. A ) ) )
22	20, 21	mpbiri 241	. . . . . . . . . 10 |- ( u = ( v i^i ( A X. A ) ) -> u C_ ( A X. A ) )
23		vex 3034	. . . . . . . . . . 11 |- u e. _V
24	23	elpw 3948	. . . . . . . . . 10 |- ( u e. ~P ( A X. A ) <-> u C_ ( A X. A ) )
25	22, 24	sylibr 217	. . . . . . . . 9 |- ( u = ( v i^i ( A X. A ) ) -> u e. ~P ( A X. A ) )
26	25	rexlimivw 2869	. . . . . . . 8 |- ( E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) -> u e. ~P ( A X. A ) )
27	26	adantl 473	. . . . . . 7 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) ) -> u e. ~P ( A X. A ) )
28		nfv 1769	. . . . . . . . . . . 12 |- F/ a ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) )
29		nfmpt1 4485	. . . . . . . . . . . . . 14 |- F/_ a ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
30	29	nfrn 5083	. . . . . . . . . . . . 13 |- F/_ a ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
31	30	nfcri 2606	. . . . . . . . . . . 12 |- F/ a w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
32	28, 31	nfan 2031	. . . . . . . . . . 11 |- F/ a ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) )
33		nfv 1769	. . . . . . . . . . 11 |- F/ a w C_ v
34	32, 33	nfan 2031	. . . . . . . . . 10 |- F/ a ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v )
35		nfmpt1 4485	. . . . . . . . . . . . 13 |- F/_ a ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) )
36	35	nfrn 5083	. . . . . . . . . . . 12 |- F/_ a ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) )
37		nfcv 2612	. . . . . . . . . . . 12 |- F/_ a ~P u
38	36, 37	nfin 3630	. . . . . . . . . . 11 |- F/_ a ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u )
39		nfcv 2612	. . . . . . . . . . 11 |- F/_ a (/)
40	38, 39	nfne 2742	. . . . . . . . . 10 |- F/ a ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/)
41		simplr 770	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> a e. RR+ )
42		ineq1 3618	. . . . . . . . . . . . . . 15 |- ( w = ( `' D " ( 0 [,) a ) ) -> ( w i^i ( A X. A ) ) = ( ( `' D " ( 0 [,) a ) ) i^i ( A X. A ) ) )
43	42	adantl 473	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( w i^i ( A X. A ) ) = ( ( `' D " ( 0 [,) a ) ) i^i ( A X. A ) ) )
44		simp2 1031	. . . . . . . . . . . . . . . 16 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> D e. (PsMet ` X ) )
45		psmetf 21400	. . . . . . . . . . . . . . . 16 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
46		ffun 5742	. . . . . . . . . . . . . . . 16 |- ( D : ( X X. X ) --> RR* -> Fun D )
47		respreima 6024	. . . . . . . . . . . . . . . 16 |- ( Fun D -> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) = ( ( `' D " ( 0 [,) a ) ) i^i ( A X. A ) ) )
48	44, 45, 46, 47	4syl 19	. . . . . . . . . . . . . . 15 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) = ( ( `' D " ( 0 [,) a ) ) i^i ( A X. A ) ) )
49	48	ad6antr 750	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) = ( ( `' D " ( 0 [,) a ) ) i^i ( A X. A ) ) )
50	43, 49	eqtr4d 2508	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( w i^i ( A X. A ) ) = ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) )
51		rspe 2844	. . . . . . . . . . . . 13 |- ( ( a e. RR+ /\ ( w i^i ( A X. A ) ) = ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) -> E. a e. RR+ ( w i^i ( A X. A ) ) = ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) )
52	41, 50, 51	syl2anc 673	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> E. a e. RR+ ( w i^i ( A X. A ) ) = ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) )
53		vex 3034	. . . . . . . . . . . . . 14 |- w e. _V
54	53	inex1 4537	. . . . . . . . . . . . 13 |- ( w i^i ( A X. A ) ) e. _V
55		eqid 2471	. . . . . . . . . . . . . 14 |- ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) = ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) )
56	55	elrnmpt 5087	. . . . . . . . . . . . 13 |- ( ( w i^i ( A X. A ) ) e. _V -> ( ( w i^i ( A X. A ) ) e. ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) <-> E. a e. RR+ ( w i^i ( A X. A ) ) = ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) )
57	54, 56	ax-mp 5	. . . . . . . . . . . 12 |- ( ( w i^i ( A X. A ) ) e. ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) <-> E. a e. RR+ ( w i^i ( A X. A ) ) = ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) )
58	52, 57	sylibr 217	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( w i^i ( A X. A ) ) e. ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) )
59		simpllr 777	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> w C_ v )
60		ssinss1 3651	. . . . . . . . . . . . 13 |- ( w C_ v -> ( w i^i ( A X. A ) ) C_ v )
61	59, 60	syl 17	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( w i^i ( A X. A ) ) C_ v )
62		inss2 3644	. . . . . . . . . . . . 13 |- ( w i^i ( A X. A ) ) C_ ( A X. A )
63	62	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( w i^i ( A X. A ) ) C_ ( A X. A ) )
64		pweq 3945	. . . . . . . . . . . . . . . 16 |- ( u = ( v i^i ( A X. A ) ) -> ~P u = ~P ( v i^i ( A X. A ) ) )
65	64	eleq2d 2534	. . . . . . . . . . . . . . 15 |- ( u = ( v i^i ( A X. A ) ) -> ( ( w i^i ( A X. A ) ) e. ~P u <-> ( w i^i ( A X. A ) ) e. ~P ( v i^i ( A X. A ) ) ) )
66	54	elpw 3948	. . . . . . . . . . . . . . 15 |- ( ( w i^i ( A X. A ) ) e. ~P ( v i^i ( A X. A ) ) <-> ( w i^i ( A X. A ) ) C_ ( v i^i ( A X. A ) ) )
67	65, 66	syl6bb 269	. . . . . . . . . . . . . 14 |- ( u = ( v i^i ( A X. A ) ) -> ( ( w i^i ( A X. A ) ) e. ~P u <-> ( w i^i ( A X. A ) ) C_ ( v i^i ( A X. A ) ) ) )
68		ssin 3645	. . . . . . . . . . . . . 14 |- ( ( ( w i^i ( A X. A ) ) C_ v /\ ( w i^i ( A X. A ) ) C_ ( A X. A ) ) <-> ( w i^i ( A X. A ) ) C_ ( v i^i ( A X. A ) ) )
69	67, 68	syl6bbr 271	. . . . . . . . . . . . 13 |- ( u = ( v i^i ( A X. A ) ) -> ( ( w i^i ( A X. A ) ) e. ~P u <-> ( ( w i^i ( A X. A ) ) C_ v /\ ( w i^i ( A X. A ) ) C_ ( A X. A ) ) ) )
70	69	ad5antlr 749	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( ( w i^i ( A X. A ) ) e. ~P u <-> ( ( w i^i ( A X. A ) ) C_ v /\ ( w i^i ( A X. A ) ) C_ ( A X. A ) ) ) )
71	61, 63, 70	mpbir2and 936	. . . . . . . . . . 11 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( w i^i ( A X. A ) ) e. ~P u )
72		inelcm 3823	. . . . . . . . . . 11 |- ( ( ( w i^i ( A X. A ) ) e. ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) /\ ( w i^i ( A X. A ) ) e. ~P u ) -> ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) )
73	58, 71, 72	syl2anc 673	. . . . . . . . . 10 |- ( ( ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) /\ a e. RR+ ) /\ w = ( `' D " ( 0 [,) a ) ) ) -> ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) )
74		simplr 770	. . . . . . . . . . 11 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) -> w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) )
75		eqid 2471	. . . . . . . . . . . . 13 |- ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) = ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
76	75	elrnmpt 5087	. . . . . . . . . . . 12 |- ( w e. _V -> ( w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. a e. RR+ w = ( `' D " ( 0 [,) a ) ) ) )
77	53, 76	ax-mp 5	. . . . . . . . . . 11 |- ( w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. a e. RR+ w = ( `' D " ( 0 [,) a ) ) )
78	74, 77	sylib 201	. . . . . . . . . 10 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) -> E. a e. RR+ w = ( `' D " ( 0 [,) a ) ) )
79	34, 40, 73, 78	r19.29af2 2915	. . . . . . . . 9 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) /\ w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) ) /\ w C_ v ) -> ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) )
80		ssn0 3770	. . . . . . . . . . . . . 14 |- ( ( A C_ X /\ A =/= (/) ) -> X =/= (/) )
81	80	ancoms 460	. . . . . . . . . . . . 13 |- ( ( A =/= (/) /\ A C_ X ) -> X =/= (/) )
82	81	3adant2 1049	. . . . . . . . . . . 12 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> X =/= (/) )
83		metuel 21657	. . . . . . . . . . . 12 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( v e. (metUnif ` D ) <-> ( v C_ ( X X. X ) /\ E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ v ) ) )
84	82, 44, 83	syl2anc 673	. . . . . . . . . . 11 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( v e. (metUnif ` D ) <-> ( v C_ ( X X. X ) /\ E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ v ) ) )
85	84	simplbda 636	. . . . . . . . . 10 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) -> E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ v )
86	85	adantr 472	. . . . . . . . 9 |- ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) -> E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ v )
87	79, 86	r19.29a 2918	. . . . . . . 8 |- ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ v e. (metUnif ` D ) ) /\ u = ( v i^i ( A X. A ) ) ) -> ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) )
88	87	r19.29an 2917	. . . . . . 7 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) ) -> ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) )
89	27, 88	jca 541	. . . . . 6 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) ) -> ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) )
90		simprl 772	. . . . . . . . . . 11 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> u e. ~P ( A X. A ) )
91	90	elpwid 3952	. . . . . . . . . 10 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> u C_ ( A X. A ) )
92		simpl3 1035	. . . . . . . . . . 11 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> A C_ X )
93		xpss12 4945	. . . . . . . . . . 11 |- ( ( A C_ X /\ A C_ X ) -> ( A X. A ) C_ ( X X. X ) )
94	92, 92, 93	syl2anc 673	. . . . . . . . . 10 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> ( A X. A ) C_ ( X X. X ) )
95	91, 94	sstrd 3428	. . . . . . . . 9 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> u C_ ( X X. X ) )
96		difssd 3550	. . . . . . . . 9 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> ( ( X X. X ) \ ( A X. A ) ) C_ ( X X. X ) )
97	95, 96	unssd 3601	. . . . . . . 8 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> ( u u. ( ( X X. X ) \ ( A X. A ) ) ) C_ ( X X. X ) )
98		simplr 770	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> b e. RR+ )
99		eqidd 2472	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) b ) ) )
100	4	imaeq2d 5174	. . . . . . . . . . . . . 14 |- ( a = b -> ( `' D " ( 0 [,) a ) ) = ( `' D " ( 0 [,) b ) ) )
101	100	eqeq2d 2481	. . . . . . . . . . . . 13 |- ( a = b -> ( ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) a ) ) <-> ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) b ) ) ) )
102	101	rspcev 3136	. . . . . . . . . . . 12 |- ( ( b e. RR+ /\ ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) b ) ) ) -> E. a e. RR+ ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) a ) ) )
103	98, 99, 102	syl2anc 673	. . . . . . . . . . 11 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> E. a e. RR+ ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) a ) ) )
104	44	ad4antr 746	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> D e. (PsMet ` X ) )
105		cnvexg 6758	. . . . . . . . . . . 12 |- ( D e. (PsMet ` X ) -> `' D e. _V )
106		imaexg 6749	. . . . . . . . . . . 12 |- ( `' D e. _V -> ( `' D " ( 0 [,) b ) ) e. _V )
107	75	elrnmpt 5087	. . . . . . . . . . . 12 |- ( ( `' D " ( 0 [,) b ) ) e. _V -> ( ( `' D " ( 0 [,) b ) ) e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. a e. RR+ ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) a ) ) ) )
108	104, 105, 106, 107	4syl 19	. . . . . . . . . . 11 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( ( `' D " ( 0 [,) b ) ) e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) <-> E. a e. RR+ ( `' D " ( 0 [,) b ) ) = ( `' D " ( 0 [,) a ) ) ) )
109	103, 108	mpbird 240	. . . . . . . . . 10 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( `' D " ( 0 [,) b ) ) e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) )
110		cnvimass 5194	. . . . . . . . . . . . . . . 16 |- ( `' D " ( 0 [,) b ) ) C_ dom D
111		fdm 5745	. . . . . . . . . . . . . . . . 17 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
112	45, 111	syl 17	. . . . . . . . . . . . . . . 16 |- ( D e. (PsMet ` X ) -> dom D = ( X X. X ) )
113	110, 112	syl5sseq 3466	. . . . . . . . . . . . . . 15 |- ( D e. (PsMet ` X ) -> ( `' D " ( 0 [,) b ) ) C_ ( X X. X ) )
114	104, 113	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( `' D " ( 0 [,) b ) ) C_ ( X X. X ) )
115		ssdif0 3741	. . . . . . . . . . . . . 14 |- ( ( `' D " ( 0 [,) b ) ) C_ ( X X. X ) <-> ( ( `' D " ( 0 [,) b ) ) \ ( X X. X ) ) = (/) )
116	114, 115	sylib 201	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( ( `' D " ( 0 [,) b ) ) \ ( X X. X ) ) = (/) )
117		0ss 3766	. . . . . . . . . . . . 13 |- (/) C_ u
118	116, 117	syl6eqss 3468	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( ( `' D " ( 0 [,) b ) ) \ ( X X. X ) ) C_ u )
119		respreima 6024	. . . . . . . . . . . . . 14 |- ( Fun D -> ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) = ( ( `' D " ( 0 [,) b ) ) i^i ( A X. A ) ) )
120	104, 45, 46, 119	4syl 19	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) = ( ( `' D " ( 0 [,) b ) ) i^i ( A X. A ) ) )
121		simpr 468	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) )
122		simpllr 777	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> v e. ~P u )
123	122	elpwid 3952	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> v C_ u )
124	121, 123	eqsstr3d 3453	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) C_ u )
125	120, 124	eqsstr3d 3453	. . . . . . . . . . . 12 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( ( `' D " ( 0 [,) b ) ) i^i ( A X. A ) ) C_ u )
126	118, 125	unssd 3601	. . . . . . . . . . 11 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( ( ( `' D " ( 0 [,) b ) ) \ ( X X. X ) ) u. ( ( `' D " ( 0 [,) b ) ) i^i ( A X. A ) ) ) C_ u )
127		ssundif 3842	. . . . . . . . . . . 12 |- ( ( `' D " ( 0 [,) b ) ) C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) <-> ( ( `' D " ( 0 [,) b ) ) \ u ) C_ ( ( X X. X ) \ ( A X. A ) ) )
128		difcom 3843	. . . . . . . . . . . 12 |- ( ( ( `' D " ( 0 [,) b ) ) \ u ) C_ ( ( X X. X ) \ ( A X. A ) ) <-> ( ( `' D " ( 0 [,) b ) ) \ ( ( X X. X ) \ ( A X. A ) ) ) C_ u )
129		difdif2 3691	. . . . . . . . . . . . 13 |- ( ( `' D " ( 0 [,) b ) ) \ ( ( X X. X ) \ ( A X. A ) ) ) = ( ( ( `' D " ( 0 [,) b ) ) \ ( X X. X ) ) u. ( ( `' D " ( 0 [,) b ) ) i^i ( A X. A ) ) )
130	129	sseq1i 3442	. . . . . . . . . . . 12 |- ( ( ( `' D " ( 0 [,) b ) ) \ ( ( X X. X ) \ ( A X. A ) ) ) C_ u <-> ( ( ( `' D " ( 0 [,) b ) ) \ ( X X. X ) ) u. ( ( `' D " ( 0 [,) b ) ) i^i ( A X. A ) ) ) C_ u )
131	127, 128, 130	3bitri 279	. . . . . . . . . . 11 |- ( ( `' D " ( 0 [,) b ) ) C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) <-> ( ( ( `' D " ( 0 [,) b ) ) \ ( X X. X ) ) u. ( ( `' D " ( 0 [,) b ) ) i^i ( A X. A ) ) ) C_ u )
132	126, 131	sylibr 217	. . . . . . . . . 10 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> ( `' D " ( 0 [,) b ) ) C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) )
133		sseq1 3439	. . . . . . . . . . 11 |- ( w = ( `' D " ( 0 [,) b ) ) -> ( w C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) <-> ( `' D " ( 0 [,) b ) ) C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) ) )
134	133	rspcev 3136	. . . . . . . . . 10 |- ( ( ( `' D " ( 0 [,) b ) ) e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) /\ ( `' D " ( 0 [,) b ) ) C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) ) -> E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) )
135	109, 132, 134	syl2anc 673	. . . . . . . . 9 |- ( ( ( ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) /\ v e. ~P u ) /\ b e. RR+ ) /\ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) -> E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) )
136		elin 3608	. . . . . . . . . . . . . 14 |- ( v e. ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) <-> ( v e. ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) /\ v e. ~P u ) )
137		vex 3034	. . . . . . . . . . . . . . . 16 |- v e. _V
138	6	elrnmpt 5087	. . . . . . . . . . . . . . . 16 |- ( v e. _V -> ( v e. ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) <-> E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) )
139	137, 138	ax-mp 5	. . . . . . . . . . . . . . 15 |- ( v e. ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) <-> E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) )
140	139	anbi1i 709	. . . . . . . . . . . . . 14 |- ( ( v e. ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) /\ v e. ~P u ) <-> ( E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) /\ v e. ~P u ) )
141		ancom 457	. . . . . . . . . . . . . 14 |- ( ( E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) /\ v e. ~P u ) <-> ( v e. ~P u /\ E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) )
142	136, 140, 141	3bitri 279	. . . . . . . . . . . . 13 |- ( v e. ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) <-> ( v e. ~P u /\ E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) )
143	142	exbii 1726	. . . . . . . . . . . 12 |- ( E. v v e. ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) <-> E. v ( v e. ~P u /\ E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) )
144		n0 3732	. . . . . . . . . . . 12 |- ( ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) <-> E. v v e. ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) )
145		df-rex 2762	. . . . . . . . . . . 12 |- ( E. v e. ~P u E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) <-> E. v ( v e. ~P u /\ E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) ) )
146	143, 144, 145	3bitr4i 285	. . . . . . . . . . 11 |- ( ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) <-> E. v e. ~P u E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) )
147	146	biimpi 199	. . . . . . . . . 10 |- ( ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) -> E. v e. ~P u E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) )
148	147	ad2antll 743	. . . . . . . . 9 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> E. v e. ~P u E. b e. RR+ v = ( `' ( D |` ( A X. A ) ) " ( 0 [,) b ) ) )
149	135, 148	r19.29vva 2920	. . . . . . . 8 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) )
150	82	adantr 472	. . . . . . . . 9 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> X =/= (/) )
151	44	adantr 472	. . . . . . . . 9 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> D e. (PsMet ` X ) )
152		metuel 21657	. . . . . . . . 9 |- ( ( X =/= (/) /\ D e. (PsMet ` X ) ) -> ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) e. (metUnif ` D ) <-> ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) C_ ( X X. X ) /\ E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) ) ) )
153	150, 151, 152	syl2anc 673	. . . . . . . 8 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) e. (metUnif ` D ) <-> ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) C_ ( X X. X ) /\ E. w e. ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) ) w C_ ( u u. ( ( X X. X ) \ ( A X. A ) ) ) ) ) )
154	97, 149, 153	mpbir2and 936	. . . . . . 7 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> ( u u. ( ( X X. X ) \ ( A X. A ) ) ) e. (metUnif ` D ) )
155		indir 3682	. . . . . . . . 9 |- ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) i^i ( A X. A ) ) = ( ( u i^i ( A X. A ) ) u. ( ( ( X X. X ) \ ( A X. A ) ) i^i ( A X. A ) ) )
156		incom 3616	. . . . . . . . . . 11 |- ( ( A X. A ) i^i ( ( X X. X ) \ ( A X. A ) ) ) = ( ( ( X X. X ) \ ( A X. A ) ) i^i ( A X. A ) )
157		disjdif 3830	. . . . . . . . . . 11 |- ( ( A X. A ) i^i ( ( X X. X ) \ ( A X. A ) ) ) = (/)
158	156, 157	eqtr3i 2495	. . . . . . . . . 10 |- ( ( ( X X. X ) \ ( A X. A ) ) i^i ( A X. A ) ) = (/)
159	158	uneq2i 3576	. . . . . . . . 9 |- ( ( u i^i ( A X. A ) ) u. ( ( ( X X. X ) \ ( A X. A ) ) i^i ( A X. A ) ) ) = ( ( u i^i ( A X. A ) ) u. (/) )
160		un0 3762	. . . . . . . . 9 |- ( ( u i^i ( A X. A ) ) u. (/) ) = ( u i^i ( A X. A ) )
161	155, 159, 160	3eqtri 2497	. . . . . . . 8 |- ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) i^i ( A X. A ) ) = ( u i^i ( A X. A ) )
162		df-ss 3404	. . . . . . . . 9 |- ( u C_ ( A X. A ) <-> ( u i^i ( A X. A ) ) = u )
163	91, 162	sylib 201	. . . . . . . 8 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> ( u i^i ( A X. A ) ) = u )
164	161, 163	syl5req 2518	. . . . . . 7 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> u = ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) i^i ( A X. A ) ) )
165		ineq1 3618	. . . . . . . . 9 |- ( v = ( u u. ( ( X X. X ) \ ( A X. A ) ) ) -> ( v i^i ( A X. A ) ) = ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) i^i ( A X. A ) ) )
166	165	eqeq2d 2481	. . . . . . . 8 |- ( v = ( u u. ( ( X X. X ) \ ( A X. A ) ) ) -> ( u = ( v i^i ( A X. A ) ) <-> u = ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) i^i ( A X. A ) ) ) )
167	166	rspcev 3136	. . . . . . 7 |- ( ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) e. (metUnif ` D ) /\ u = ( ( u u. ( ( X X. X ) \ ( A X. A ) ) ) i^i ( A X. A ) ) ) -> E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) )
168	154, 164, 167	syl2anc 673	. . . . . 6 |- ( ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) /\ ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) -> E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) )
169	89, 168	impbida 850	. . . . 5 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) <-> ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) ) )
170		eqid 2471	. . . . . . 7 |- ( v e. (metUnif ` D ) |-> ( v i^i ( A X. A ) ) ) = ( v e. (metUnif ` D ) |-> ( v i^i ( A X. A ) ) )
171	170	elrnmpt 5087	. . . . . 6 |- ( u e. _V -> ( u e. ran ( v e. (metUnif ` D ) |-> ( v i^i ( A X. A ) ) ) <-> E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) ) )
172	23, 171	ax-mp 5	. . . . 5 |- ( u e. ran ( v e. (metUnif ` D ) |-> ( v i^i ( A X. A ) ) ) <-> E. v e. (metUnif ` D ) u = ( v i^i ( A X. A ) ) )
173		pweq 3945	. . . . . . . 8 |- ( v = u -> ~P v = ~P u )
174	173	ineq2d 3625	. . . . . . 7 |- ( v = u -> ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P v ) = ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) )
175	174	neeq1d 2702	. . . . . 6 |- ( v = u -> ( ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P v ) =/= (/) <-> ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) )
176	175	elrab 3184	. . . . 5 |- ( u e. { v e. ~P ( A X. A ) | ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P v ) =/= (/) } <-> ( u e. ~P ( A X. A ) /\ ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P u ) =/= (/) ) )
177	169, 172, 176	3bitr4g 296	. . . 4 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( u e. ran ( v e. (metUnif ` D ) |-> ( v i^i ( A X. A ) ) ) <-> u e. { v e. ~P ( A X. A ) | ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P v ) =/= (/) } ) )
178	177	eqrdv 2469	. . 3 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ran ( v e. (metUnif ` D ) |-> ( v i^i ( A X. A ) ) ) = { v e. ~P ( A X. A ) | ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P v ) =/= (/) } )
179	19, 178	eqtrd 2505	. 2 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( (metUnif ` D ) ↾t ( A X. A ) ) = { v e. ~P ( A X. A ) | ( ran ( a e. RR+ |-> ( `' ( D |` ( A X. A ) ) " ( 0 [,) a ) ) ) i^i ~P v ) =/= (/) } )
180	11, 13, 179	3eqtr4rd 2516	1 |- ( ( A =/= (/) /\ D e. (PsMet ` X ) /\ A C_ X ) -> ( (metUnif ` D ) ↾t ( A X. A ) ) = (metUnif ` ( D |` ( A X. A ) ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   E.wrex 2757   {crab 2760   _Vcvv 3031   \ cdif 3387   u. cun 3388   i^i cin 3389   C_ wss 3390   (/)c0 3722   ~Pcpw 3942   |-> cmpt 4454   X. cxp 4837   `'ccnv 4838   dom cdm 4839   ran crn 4840   |` cres 4841   "cima 4842   Fun wfun 5583   -->wf 5585   ` cfv 5589  (class class class)co 6308   0cc0 9557   RR*cxr 9692   RR+crp 11325   [,)cico 11662   ↾_t crest 15397  PsMetcpsmet 19031   fBascfbas 19035   filGencfg 19036  metUnifcmetu 19038

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-rep 4508  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

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-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  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-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-1st 6812  df-2nd 6813  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-rp 11326  df-ico 11666  df-rest 15399  df-psmet 19039  df-fbas 19044  df-fg 19045  df-metu 19046

This theorem is referenced by:  reust  22418  qqhucn  28870