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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.
StepHypRef Expression
1 relres 5132 . . 3  |-  Rel  (  _I  |`  X )
21a1i 11 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  Rel  (  _I  |`  X ) )
3 vex 3048 . . . . . . . . . . . . . . 15  |-  q  e. 
_V
43brres 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 ) )
63ideq 4987 . . . . . . . . . . . . . . 15  |-  ( p  _I  q  <->  p  =  q )
76anbi1i 701 . . . . . . . . . . . . . 14  |-  ( ( p  _I  q  /\  p  e.  X )  <->  ( p  =  q  /\  p  e.  X )
)
84, 5, 73bitr3i 279 . . . . . . . . . . . . 13  |-  ( <.
p ,  q >.  e.  (  _I  |`  X )  <-> 
( p  =  q  /\  p  e.  X
) )
98biimpi 198 . . . . . . . . . . . 12  |-  ( <.
p ,  q >.  e.  (  _I  |`  X )  ->  ( p  =  q  /\  p  e.  X ) )
109ad2antlr 733 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( p  =  q  /\  p  e.  X ) )
1110simpld 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 >. )
1413fveq2d 5869 . . . . . . . . . . 11  |-  ( p  =  q  ->  ( D `  <. p ,  p >. )  =  ( D `  <. p ,  q >. )
)
1512, 14syl5eq 2497 . . . . . . . . . 10  |-  ( p  =  q  ->  (
p D p )  =  ( D `  <. p ,  q >.
) )
1611, 15syl 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 ) )
1810simprd 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 )
2017, 18, 19syl2anc 667 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  ( p D p )  =  0 )
2116, 20eqtr3d 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*
2322a1i 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
) )
2723, 24, 25, 26syl3anc 1268 . . . . . . . . 9  |-  ( a  e.  RR+  ->  0  e.  ( 0 [,) a
) )
2827adantl 468 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  0  e.  ( 0 [,) a ) )
2921, 28eqeltrd 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 )
3230, 31syl 17 . . . . . . . . 9  |-  ( D  e.  (PsMet `  X
)  ->  Fun  D )
3332ad3antrrr 736 . . . . . . . 8  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  Fun  D )
3411, 18eqeltrrd 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
) )
3618, 34, 35syl2anc 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 ) )
3830, 37syl 17 . . . . . . . . . 10  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
3938ad3antrrr 736 . . . . . . . . 9  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  dom  D  =  ( X  X.  X
) )
4036, 39eleqtrrd 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
) ) ) )
4233, 40, 41syl2anc 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
) ) ) )
4329, 42mpbid 214 . . . . . 6  |-  ( ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F
)  /\  <. p ,  q >.  e.  (  _I  |`  X ) )  /\  a  e.  RR+ )  ->  <. p ,  q
>.  e.  ( `' D " ( 0 [,) a
) ) )
4443adantr 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 ) ) )
4644, 45eleqtrrd 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
) ) )
4948metustel 21565 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
5049ad2antrr 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
) ) ) )
5147, 50mpbid 214 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
5246, 51r19.29a 2932 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  <. p ,  q >.  e.  A
)
5352ex 436 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  ( <. p ,  q >.  e.  (  _I  |`  X )  ->  <. p ,  q
>.  e.  A ) )
542, 53relssdv 4927 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  (  _I  |`  X )  C_  A )
Colors of variables: wff setvar class
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
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