Theorem metustid 22359

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 5426	. . 3 |- Rel ( _I |` X )
2	1	a1i 11	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> Rel ( _I |` X ) )
3		vex 3203	. . . . . . . . . . . . . . 15 |- q e. _V
4	3	brres 5402	. . . . . . . . . . . . . 14 |- ( p ( _I |` X ) q <-> ( p _I q /\ p e. X ) )
5		df-br 4654	. . . . . . . . . . . . . 14 |- ( p ( _I |` X ) q <-> <. p , q >. e. ( _I |` X ) )
6	3	ideq 5274	. . . . . . . . . . . . . . 15 |- ( p _I q <-> p = q )
7	6	anbi1i 731	. . . . . . . . . . . . . 14 |- ( ( p _I q /\ p e. X ) <-> ( p = q /\ p e. X ) )
8	4, 5, 7	3bitr3i 290	. . . . . . . . . . . . 13 |- ( <. p , q >. e. ( _I |` X ) <-> ( p = q /\ p e. X ) )
9	8	biimpi 206	. . . . . . . . . . . 12 |- ( <. p , q >. e. ( _I |` X ) -> ( p = q /\ p e. X ) )
10	9	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p = q /\ p e. X ) )
11	10	simpld 475	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> p = q )
12		df-ov 6653	. . . . . . . . . . 11 |- ( p D p ) = ( D ` <. p , p >. )
13		opeq2 4403	. . . . . . . . . . . 12 |- ( p = q -> <. p , p >. = <. p , q >. )
14	13	fveq2d 6195	. . . . . . . . . . 11 |- ( p = q -> ( D ` <. p , p >. ) = ( D ` <. p , q >. ) )
15	12, 14	syl5eq 2668	. . . . . . . . . 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 798	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> D e. (PsMet ` X ) )
18	10	simprd 479	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> p e. X )
19		psmet0 22113	. . . . . . . . . 10 |- ( ( D e. (PsMet ` X ) /\ p e. X ) -> ( p D p ) = 0 )
20	17, 18, 19	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( p D p ) = 0 )
21	16, 20	eqtr3d 2658	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( D ` <. p , q >. ) = 0 )
22		0xr 10086	. . . . . . . . . . 11 |- 0 e. RR*
23	22	a1i 11	. . . . . . . . . 10 |- ( a e. RR+ -> 0 e. RR* )
24		rpxr 11840	. . . . . . . . . 10 |- ( a e. RR+ -> a e. RR* )
25		rpgt0 11844	. . . . . . . . . 10 |- ( a e. RR+ -> 0 < a )
26		lbico1 12228	. . . . . . . . . 10 |- ( ( 0 e. RR* /\ a e. RR* /\ 0 < a ) -> 0 e. ( 0 [,) a ) )
27	23, 24, 25, 26	syl3anc 1326	. . . . . . . . 9 |- ( a e. RR+ -> 0 e. ( 0 [,) a ) )
28	27	adantl 482	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> 0 e. ( 0 [,) a ) )
29	21, 28	eqeltrd 2701	. . . . . . 7 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> ( D ` <. p , q >. ) e. ( 0 [,) a ) )
30		psmetf 22111	. . . . . . . . . 10 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
31		ffun 6048	. . . . . . . . . 10 |- ( D : ( X X. X ) --> RR* -> Fun D )
32	30, 31	syl 17	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> Fun D )
33	32	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> Fun D )
34	11, 18	eqeltrrd 2702	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> q e. X )
35		opelxpi 5148	. . . . . . . . . 10 |- ( ( p e. X /\ q e. X ) -> <. p , q >. e. ( X X. X ) )
36	18, 34, 35	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. ( X X. X ) )
37		fdm 6051	. . . . . . . . . . 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 766	. . . . . . . . 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 2704	. . . . . . . 8 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) /\ a e. RR+ ) -> <. p , q >. e. dom D )
41		fvimacnv 6332	. . . . . . . 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 693	. . . . . . 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 222	. . . . . 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 481	. . . . 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 477	. . . . 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 2704	. . . 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 792	. . . . 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 22355	. . . . . 6 |- ( D e. (PsMet ` X ) -> ( A e. F <-> E. a e. RR+ A = ( `' D " ( 0 [,) a ) ) ) )
50	49	ad2antrr 762	. . . . 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 222	. . . 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 3078	. . 3 |- ( ( ( D e. (PsMet ` X ) /\ A e. F ) /\ <. p , q >. e. ( _I |` X ) ) -> <. p , q >. e. A )
53	52	ex 450	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> ( <. p , q >. e. ( _I |` X ) -> <. p , q >. e. A ) )
54	2, 53	relssdv 5212	1 |- ( ( D e. (PsMet ` X ) /\ A e. F ) -> ( _I |` X ) C_ A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Rel wrel 5119   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   < clt 10074   RR+crp 11832   [,)cico 12177  PsMetcpsmet 19730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-rp 11833  df-ico 12181  df-psmet 19738

This theorem is referenced by:  metustfbas  22362  metust  22363