Theorem metustid 21569

Description: The identity diagonal is included in all elements of the filter base generated by the metric D. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
metust.1	|- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )

Assertion

Ref	Expression
metustid	|- ( ( D e. (PsMet ` X ) /\ A e. F ) -> ( _I |` X ) C_ A )

Distinct variable groups:   D, a   X, a   A, a   F, a

Proof of Theorem metustid

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 5132	. . 3 |- Rel ( _I |` X )
2	1	a1i 11	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> Rel ( _I |` X ) )
3		vex 3048	. . . . . . . . . . . . . . 15 |- q e. _V
4	3	brres 5111	. . . . . . . . . . . . . 14 |- ( p ( _I |` X ) q <-> ( p _I q /\ p e. X ) )
5		df-br 4403	. . . . . . . . . . . . . 14 |- ( p ( _I |` X ) q <-> <. p , q >. e. ( _I |` X ) )
6	3	ideq 4987	. . . . . . . . . . . . . . 15 |- ( p _I q <-> p = q )
7	6	anbi1i 701	. . . . . . . . . . . . . 14 |- ( ( p _I q /\ p e. X ) <-> ( p = q /\ p e. X ) )
8	4, 5, 7	3bitr3i 279	. . . . . . . . . . . . 13 |- ( <. p , q >. e. ( _I |` X ) <-> ( p = q /\ p e. X ) )
9	8	biimpi 198	. . . . . . . . . . . 12 |- ( <. p , q >. e. ( _I |` X ) -> ( p = q /\ p e. X ) )
10	9	ad2antlr 733	. . . . . . . . . . 11 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p = q /\ p e. X ) )
11	10	simpld 461	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> p = q )
12		df-ov 6293	. . . . . . . . . . 11 |- ( p D p ) = ( D ` <. p , p >. )
13		opeq2 4167	. . . . . . . . . . . 12 |- ( p = q -> <. p , p >. = <. p , q >. )
14	13	fveq2d 5869	. . . . . . . . . . 11 |- ( p = q -> ( D ` <. p , p >. ) = ( D ` <. p , q >. ) )
15	12, 14	syl5eq 2497	. . . . . . . . . 10 |- ( p = q -> ( p D p ) = ( D ` <. p , q >. ) )
16	11, 15	syl 17	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p D p ) = ( D ` <. p , q >. ) )
17		simplll 768	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> D e. (PsMet ` X ) )
18	10	simprd 465	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> p e. X )
19		psmet0 21324	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ p e. X ) -> ( p D p ) = 0 )
20	17, 18, 19	syl2anc 667	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p D p ) = 0 )
21	16, 20	eqtr3d 2487	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( D ` <. p , q >. ) = 0 )
22		0xr 9687	. . . . . . . . . . 11 |- 0 e. RR*
23	22	a1i 11	. . . . . . . . . 10 |- ( a e. RR+ -> 0 e. RR* )
24		rpxr 11309	. . . . . . . . . 10 |- ( a e. RR+ -> a e. RR* )
25		rpgt0 11313	. . . . . . . . . 10 |- ( a e. RR+ -> 0 < a )
26		lbico1 11689	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ a e. RR* /\ 0 < a ) -> 0 e. ( 0 [,) a ) )
27	23, 24, 25, 26	syl3anc 1268	. . . . . . . . 9 |- ( a e. RR+ -> 0 e. ( 0 [,) a ) )
28	27	adantl 468	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> 0 e. ( 0 [,) a ) )
29	21, 28	eqeltrd 2529	. . . . . . 7 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( D ` <. p , q >. ) e. ( 0 [,) a ) )
30		psmetf 21322	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
31		ffun 5731	. . . . . . . . . 10 |- ( D : ( X X. X ) --> RR* -> Fun D )
32	30, 31	syl 17	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> Fun D )
33	32	ad3antrrr 736	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> Fun D )
34	11, 18	eqeltrrd 2530	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> q e. X )
35		opelxpi 4866	. . . . . . . . . 10 |- ( ( p e. X /\ q e. X ) -> <. p , q >. e. ( X X. X ) )
36	18, 34, 35	syl2anc 667	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. ( X X. X ) )
37		fdm 5733	. . . . . . . . . . 11 |- ( D : ( X X. X ) --> RR* -> dom D = ( X X. X ) )
38	30, 37	syl 17	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> dom D = ( X X. X ) )
39	38	ad3antrrr 736	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> dom D = ( X X. X ) )
40	36, 39	eleqtrrd 2532	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. dom D )
41		fvimacnv 5997	. . . . . . . 8 |- ( ( Fun D /\ <. p , q >. e. dom D ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
42	33, 40, 41	syl2anc 667	. . . . . . 7 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( ( D ` <. p , q >. ) e. ( 0 [,) a ) <-> <. p , q >. e. ( `' D " ( 0 [,) a ) ) ) )
43	29, 42	mpbid 214	. . . . . 6 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. ( `' D " ( 0 [,) a ) ) )
44	43	adantr 467	. . . . 5 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. p , q >. e. ( `' D " ( 0 [,) a ) ) )
45		simpr 463	. . . . 5 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> A = ( `' D " ( 0 [,) a ) ) )
46	44, 45	eleqtrrd 2532	. . . 4 |- ( ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) /\ A = ( `' D " ( 0 [,) a ) ) ) -> <. p , q >. e. A )
47		simplr 762	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> A e. F )
48		metust.1	. . . . . . 7 |- F = ran ( a e. RR+ |-> ( `' D " ( 0 [,) a ) ) )
49	48	metustel 21565	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
50	49	ad2antrr 732	. . . . 5 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
51	47, 50	mpbid 214	. . . 4 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) )
52	46, 51	r19.29a 2932	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> <. p , q >. e. A )
53	52	ex 436	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> ( <. p , q >. e. ( _I |` X ) -> <. p , q >. e. A ) )
54	2, 53	relssdv 4927	1 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> ( _I |` X ) C_ A )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   E.wrex 2738   C_ wss 3404   <.cop 3974   class class class wbr 4402   |-> cmpt 4461   _I cid 4744   X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835   |` cres 4836   "cima 4837   Rel wrel 4839   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   0cc0 9539   RR*cxr 9674   < clt 9675   RR+crp 11302   [,)cico 11637  PsMetcpsmet 18954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-i2m1 9607  ax-1ne0 9608  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-rp 11303  df-ico 11641  df-psmet 18962

This theorem is referenced by:  metustfbas  21572  metust  21573